q–Zeilberger Examples

Axel Riese

Research Institute for Symbolic Computation
J. Kepler University Linz
A–4040 Linz, Austria

Axel.Riese@risc.uni–linz.ac.at

February 21, 2005

Introduction

This Mathematica notebook for version 3 and higher accompanies the Mathematica implementation of a q–analogue of Zeilberger's algorithm written by Axel Riese. Since Mathematica notebooks are now machine independent, it replaces the old (ASCII) collection qZeilExamples.txt. The identities taken from several publications are sorted alphabetically by authors. For more details see section References below. Among these publications, the excellent book Basic Hypergeometric Series by G. Gasper and M. Rahman has served as my standard reference. Thus, if you look for specific identities, first have a look there. At the moment, this notebook contains Mathematica input for about 490 identities.

Detailed comments on the installation and use of the package (including theoretical and practical background information) can be found in the paper

Peter Paule and Axel Riese, A Mathematica q–Analogue of Zeilberger's Algorithm Based on an Algebraically Motivated Approach to q–Hypergeometric Telescoping, in "Special Functions, q–Series and Related Topics", Fields Inst. Commun., Vol. 14, pp. 179–210, 1997

and in my Ph.D. thesis

Axel Riese, Contributions to Symbolic q–Hypergeometric Summation, Ph.D. thesis, RISC, J. Kepler University Linz, 1997.

Methods for tuning the algorithm (in particular an explanation for the strange substitutions made for several examples below) are described in the paper

Axel Riese,  Fine Tuning Zeilberger's Algorithm: The Methods of Automatic Filtering and Creative Substituting, to appear in "Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics" (F.G. Garvan and M. Ismail, eds.), Developments in Mathematics, Kluwer, 2001.

Those publications (as well as the latest version of the package) are available in electronic form at the qZeil homepage

http://www.risc.uni–linz.ac.at/research/combinat/software/qZeil

A detailed list of changes since version 1.4 and known bugs is given in the accompanying file WhatsNew.txt.

All tests have been done on an SGI Octane using Mathematica 3.0.2 under IRIX 6.5. Runtimes are included for those examples which cannot be solved within approximately three minutes on this workstation.

Please report any kind of comments or criticism to

Axel Riese
Research Institute for Symbolic Computation
Johannes Kepler University
A–4040 Linz, Austria
Email address: Axel.Riese@risc.uni–linz.ac.at

Note: To run the examples for bibasic hypergeometric summation, i.e. those using the function pqTelescope[], you have to download the package Bibasic Telescope separately. It is available at

http://www.risc.uni–linz.ac.at/research/combinat/risc/software/Bibasic

Acknowledgments: Valuable contributions to the work on this package have been provided by the RISC combinatorics group. In particular I would like to thank my advisor Peter Paule (for successfully spreading the q–disease and much much more), Markus Schorn (for writing the interface LinSolve), Erhard Aichinger (for his great ENullSpace algorithm), Christian "Malles" Mallinger, and Carsten Schneider (for fruitful comments and friendship over the years), Fabrizio Caruso, Istvan Nemes, Roberto Pirastu, Kurt Wegschaider, Roman Winkler, and Burkhard Zimmermann (for discussions), and the RISC colleagues Peter Stadelmeyer (for being THE software guru), Tobias Beck, and Wolfgang Windsteiger. I also thank my parents for financially supporting this project as well as my children (Claudia, Michaela, and Alexander) for sharing pleasure and frustration with me.

Furthermore, I am specially grateful to the following persons for their cooperation and bug reports (starring in alphabetical order): George E. Andrews, Alexander Berkovich, Harald Böing, Frederic Chyzak, Johann Cigler, Stavros Garoufalidis, George Gasper, Bruno Gauthier, Wolfram Koepf, Tom H. Koornwinder, Christian Krattenthaler, Marko Petkovšek, Volker Strehl, Raimundas Vidunas, Herbert S. Wilf, Doron Zeilberger.

References

The examples in this notebook have been taken from the following publications:

W.A. Al–Salam and A. Verma [1984], On Quadratic Transformations of Basic Series, SIAM J. Math. Anal., 15, 414–421.

G.E. Andrews [1974], Applications of Basic Hypergeometric Functions, SIAM Rev., 16, 441–484.

G.E. Andrews [1976], The Theory of Partitions, Encyclopedia Math. Appl., Vol. 2, Addison–Wesley, Reading. Reissued: Cambridge University Press, London and New York, 1985.

G.E. Andrews [1981], Ramanujan's "Lost" Notebook. I. Partial Theta–Functions, Adv. Math., 41, 137–172.

G.E. Andrews [1984], Multiple Series Rogers–Ramanujan Type Identities, Pacific J. Math., 114, 267–283.

G.E. Andrews [1986a], q–Series. Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional  Conf. Ser. in Math., No. 66, Amer. Math. Soc., Providence.

G.E. Andrews [1986b], The Fifth and Seventh Order Mock Theta Functions, Trans. Amer. Math. Soc., 293, 113–134.

G.E. Andrews [1992], Bailey Chains and Generalized Lambert Series: I. Four Identities of Ramanujan, Illinois J. Math., 36, 251–274.

G.E. Andrews [1995], On a Conjecture of Peter Borwein, J. Symbolic Comput., 20, 487–501.

G.E. Andrews and D.M. Bressoud [1984], Identities in Combinatorics III, Discrete Math., 49, 223–236.

W.N. Bailey [1947],  Some Identities in Combinatory Analysis, Proc. London Math. Soc. (2), 49, 421–435.

W.N. Bailey [1949],  Identities of the Rogers–Ramanujan Type, Proc. London Math. Soc. (2), 50, 1–10.

D.M. Bressoud [1981], Some Identities for Terminating q–Series, Math. Proc. Cambbridge Philos. Soc., 89, 211–223.

D.M. Bressoud [1983], A Matrix Inverse, Proc. Amer. Math. Soc., 88, 446–448.

D.M. Bressoud [1987], Almost Poised Basic Hypergeometric Series, Proc. Indian Acad. Sci. Math. Sci., 96, 61–66.

L. Carlitz [1969], Some Formulas of F.H. Jackson, Monatsh. Math., 73, 193–198.

W. Chu [1993], Inversion Techniques and Combinatorial Identities, Bollettino U.M.I (7), 7–B, 737–760.

W. Chu [1995a], Basic Almost–Poised Hypergeometric Series, IAMI 95.3.

W. Chu [1995b],  Symmetry on q–Pfaff–Saalschuetz–Sheppard series, Pure Math. Appl., 6, 31–39.

J. Cigler [1979], Operatorenmethoden fuer q–Identitaeten, Monatsh. Math., 88, 87–105.

J. Cigler [1982], Elementare q–Identitaeten, Sem. Lothar. Combin., 5e Session, Sainte–Croix–aux Mines.

G. Gasper [1989a], Summation, Transformation, and Expansion Formulas for Bibasic Series, Trans. Amer. Math. Soc., 312, 257–277.

G. Gasper [1989b], q–Extensions of Clausen's Formula and of the Inequalities Used by de Branges in his Proof of the Bieberbach, Robertson, and Milin Conjectures, SIAM J. Math. Anal., 20, 1019–1034.

G. Gasper [1997], Elementary Derivations of Summation and Transformation Formulas for q–Series, in "Special Functions, q–Series and Related Topics", Fields Inst. Commun., Vol. 14, 55–70.

G. Gasper and M. Rahman [1990a], An Indefinite Bibasic Summation Formula and Some Quadratic, Cubic, and Quartic Summation and Transformation Formulas, Canad. J. Math., 42, 1–27.

G. Gasper and M. Rahman [1990b], Basic Hypergeometric Series, Encyclopedia Math. Appl., Vol. 35, Cambridge University Press, London and New York.

I.M. Gessel and D. Stanton [1983], Applications of q–Lagrange Inversion to Basic Hypergeometric Series, Trans. Amer. Math. Soc., 277, 173–201.

V.K. Jain [1981], Some Transformations of Basic Hypergeometric Functions II, SIAM J. Math. Anal., 12, 957–961.

V.K. Jain and A. Verma [1980], Transformations between Basic Hypergeometric Series on Different Bases and Identities of Rogers–Ramanujan Type, J. Math. Anal. Appl., 76, 230–269.

T.H. Koornwinder [1997], Special Functions and q–Commuting Variables, in "Special Functions, q–Series and Related Topics", Fields Inst. Commun., Vol. 14, 131–166.

C. Krattenthaler [1984], A New q–Lagrange Formula and Some Applications, Proc. Amer. Math. Soc., 90, 338–344.

C. Krattenthaler [1996], A New Matrix Inverse, Proc. Amer. Math. Soc., 124, 47–59.

A. Lakin [1952], A Hypergeometric Identity Related to Dougall's Theorem, J. London Math. Soc., 27, 229–234.

P. Paule [1982], Zwei neue Transformationen als elementare Anwendungen der q–Vandermonde Formel, PhD Thesis, University of Vienna.

P. Paule [1985a], On Identities of the Rogers–Ramanujan Type, J. Math. Anal. Appl., 107, 255–284.

P. Paule [1985b], Ueber das Involutionsprinzip von Garsia und Milne, Bayreuth. Math. Schr., 21, 295–319.

P. Paule [1987], The Concept of Bailey Chains, Sem. Lothar. Combin., 18e Session, Obsteig.

P. Paule [1994], Short and Easy Computer Proofs of the Rogers–Ramanujan Identities and of Identities of Similar Type, Electron. J. Combin., 1, #R10.

P. Paule and A. Riese [1997], A Mathematica q–Analogue of Zeilberger's Algorithm Based on an Algebraically Motivated Approach to q–Hypergeometric Telescoping, in "Special Functions, q–Series and Related Topics", Fields Inst. Commun., Vol. 14, 179–210.

P. Paule and V. Strehl [1995], Symbolic Summation — Some Recent Developments, in "Computeralgebra in Science and Engineering — Algorithms, Systems, Applications", World Scientific, Singapore, 138–162.

M. Rahman and A. Verma [1993], Quadratic Transformation Formulas for Basic Hypergeometric Series, Trans. Amer. Math. Soc., 335, 277–302.

L.J. Slater [1951], A New Proof of Rogers's Transformations of Infinite Series, Proc. London Math. Soc. (2), 53, 460–475.

L.J. Slater [1952], Further Identities of the Rogers–Ramanujan Type, Proc. London Math. Soc. (2), 54, 147–167.

L.J. Slater [1966], Generalized Hypergeometric Functions, Cambridge University Press, Cambridge.

H.M. Srivastava [1983], Certain q–Polynomial Expansions for Functions of Several Variables, IMA J. Appl. Math., 30, 315–323.

A. Verma [1977], Certain Summation Formulae for Basic Hypergeometric Series, Canad. Math. Bull., 20, 369–375.

H.S. Wilf and D. Zeilberger [1992], An Algorithmic Proof Theory for Hypergeometric (Ordinary and "q") Multisum/Integral Identities, Invent. Math., 108, 575–633.

Examples

Initialization Stuff

<<RISC`qZeil`

Al-Salam, Verma [1984]

(4.2) with b^2->b, c^2->c, k->n-k (corrected, LHS 220 sec)

qZeil[qPochhammer[-q^(1-n)/x,q,n] qPochhammer[q^(-2n),q^2,k] (1+q^(2k-n)) *
      qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] qPochhammer[x,q,k] q^(k(k+1)/2) *
      (q^(2-2n)/(b c x))^k /
     (qPochhammer[-q^(1-n),q,n] (1+q^(-n)) qPochhammer[q^(2-2n)/b,q^2,k] *
      qPochhammer[q^(2-2n)/c,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[-q^(1-n)/x,q,k]) /.
      k->n-k, {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[q^(2-2n)/(b c),q^2,k] qPochhammer[q^(-n),q,2k] qPochhammer[x^2,q^2,k] *
      (-q/x^2)^k q^(-2Binomial[k,2]) /
     (qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^(2-2n)/c,q^2,k] qPochhammer[q^2,q^2,k]) /.
      k->n-k, {k, -Infinity, Infinity}, n, 3]

(4.3, b=1) (inversion trick)

qZeil[qPochhammer[w/a,q,k] (1-a/w q^k) qPochhammer[a q/w,q,2n-k-1] *
      q^(2n+k(k-3)/2) (a q/w)^k qPochhammer[w,q,k] qPochhammer[-q,q,k] *
      qPochhammer[a^2 q^2,q^2,n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^2,q^2,k] qPochhammer[w/a,q,k] qPochhammer[-a q,q,k] *
      qPochhammer[a,q,2n] qPochhammer[a^2 q^2/w^2,q^2,n] q^(2n)), {k, 0, n}, n, 1]

Andrews [1974]

Th. 3.9 with z:=q^n

qZeil[(-1)^k q^(k(3k-1)/2+3n k) (1+q^(n+k)) /
     (qPochhammer[-q^(1-n),q,Infinity] qPochhammer[-q^n,q,Infinity] *
      qPochhammer[q^(1-2n),q^2,Infinity] qPochhammer[q^(1+2n),q^2,Infinity]),
      {k, -Infinity, Infinity}, n, 1]

(5.14)

qZeil[qBinomial[2n+1,2o+2k+1,q] qBinomial[o+k,k,q^2] q^(k(2o+2k+1)) /
     (qBinomial[2n-o,o,q^2] qPochhammer[-q,q,2n-2o]),
      {k, -Infinity, Infinity}, n, 1, {o}]

(5.15)

qZeil[qBinomial[2n,2o+2k,q] qBinomial[o+k,k,q^2] q^(k(2o+2k-1)) *
      (1-q^(4n-2o)) /
     (qBinomial[2n-o,o,q^2] qPochhammer[-q,q,2n-2o] (1-q^(2n))),
      {k, -Infinity, Infinity}, n, 1, {o}]

(5.17) with n->2n

qZeil[(-1)^k qBinomial[2n,k,q] qBinomial[2k,k,q] *
      q^((2n-k)(2n-k-1)/2) qPochhammer[-q,q,n]^2 /
      (qPochhammer[-q,q,k] qBinomial[2n,n,q] q^(2n^2)), {k, 0, 2n}, n, 1]

Andrews [1976]

Ex. 2.2 with b:=q^(-n)

qZeil[qPochhammer[q^(-n),q,2k] t^(2k) / qPochhammer[q^2,q^2,k], {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[-t q^(-n),q,n] qPochhammer[q^(-n),q,k] t^k /
     (qPochhammer[q,q,k] qPochhammer[-t q^(-n),q,k]), {k, 0, n}, n, 2]

Ex. 2.3 with t:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q^2,k] b^k / qPochhammer[q,q,k], {k, 0, n}, n, 2]

qZeil[qPochhammer[b q^(-2n+1),q^2,n] qPochhammer[q^(-2n),q^2,k] b^k /
     (qPochhammer[q^2,q^2,k] qPochhammer[b q^(-2n+1),q^2,k]), {k, 0, n}, n, 2]

Ex. 2.4 with b:=q^(-2n)

qZeil[qPochhammer[a,q,k] qPochhammer[q^(-2n),q^2,k] t^k /
     (qPochhammer[q,q,k] qPochhammer[a t q^(-2n),q^2,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[t q^(-2n),q^2,n] qPochhammer[a,q^2,k] qPochhammer[q^(-2n),q^2,k] (t q)^k /
     (qPochhammer[a t q^(-2n),q^2,n] qPochhammer[q^2,q^2,k] qPochhammer[t q^(-2n),q^2,k]),
      {k, 0, n}, n, 2]

Ex. 2.5 with a:=q^n

qZeil[(1+q^n) (-q^n)^k / qPochhammer[-q,q,k], {k, 0, Infinity}, n, 1]

qZeil[q^(k^2+n k) / (qPochhammer[-q,q,k] qPochhammer[-q^(1+n),q,k]),
      {k, 0, Infinity}, n, 1]

Ex. 2.6 with x:=q^n

qZeil[q^(k^2+2n k) / qPochhammer[y,q^2,k+1], {k, 0, Infinity}, n, 1]

qZeil[qPochhammer[-q^(1+2n)/y,q^2,k] y^k, {k, 0, Infinity}, n, 1]

Ex. 2.15 with z:=q^n

qZeil[q^(n k) / qPochhammer[-q^n,q,k+1], {k, 0, Infinity}, n, 1]

qZeil[qPochhammer[q,q^2,k] q^(2n k), {k, 0, Infinity}, n, 1]

(3.3.6)

qZeil[qBinomial[n,k,q] (-z)^k q^(k(k-1)/2) / qPochhammer[z,q,n],
      {k, 0, n}, n, 1]

(3.3.9)

qZeil[qBinomial[m+k,k,q] q^k, {k, 0, n}, n, 0, {m}]

(3.3.10)

qZeil[qBinomial[a,k,q] qBinomial[b,n-k,q] q^((a-k)(n-k)) /
      qBinomial[a+b,n,q], {k, 0, n}, n, 1, {a, b}]

Ex. 3.1

qZeil[qBinomial[2n,n+k,q^2] q^(k^2) x^k /
     (qPochhammer[-q/x,q^2,n] qPochhammer[-x q,q^2,n]), {k, -n, n}, n, 1]

(9.2.3) with x:=q^n

qZeil[(-1)^k q^(k(3k+1)/2+n k) (1-q^(n+2k+1)) /
     (qPochhammer[q,q,k] qPochhammer[q^(n+k+1),q,Infinity]), {k, 0, Infinity}, n, 1]

Andrews [1981]

(4.3) with n->2m-n (LHS)

qZeil[qPochhammer[q,q^2,m] (-1)^n / qPochhammer[q,q,2m-n], {n, 0, 2m}, m, 1]

qZeil[(-1)^n q^(n^2) / qPochhammer[q^2,q^2,m-n], {n, 0, m}, m, 1]

(5.1)

qZeil[qBinomial[m,n,q] qPochhammer[q^(m+1),q,n] (-1)^n *
      q^(Binomial[n+1,2]-m n)/.n->n+m-i, {n, 0, i}, i, 1, {m}]

qZeil[(-1)^(m+i) qPochhammer[q^(m+1),q,m-i] q^(-Binomial[m,2]+
      Binomial[i+1,2]) qBinomial[m-j-1,i-j,q] q^(2m j-i j),
      {j, 0, i}, i, 1, {m}]

(5.2)

qZeil[qPochhammer[q,q,n+m] (-1)^n q^(Binomial[n+1,2]-m n) /
     (qPochhammer[q,q,n] qPochhammer[q,q,m-n] q^(m^2+m)), {n, 0, m}, m, 1]

Andrews [1984]

(2.14)

qZeil[(1-a q^(2j)) qPochhammer[q^(-n),q,j] qPochhammer[a,q,j] q^(n j) /
     ((1-a) qPochhammer[a q^(n+1),q,j] qPochhammer[q,q,j]),
      {j, 0, m}, n, 0]

Andrews [1986a]

(3.37)

qZeil[(-1)^k q^(k(k-1)/2) (1-q^(2k+1)) /
     ((1-q) qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k]), {k, 0, n}, n, 0]

(3.40)

qZeil[(1-a q^(2n)) qPochhammer[a q,q,n+k-1] (-1)^(n-k) q^Binomial[n-k,2] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,k-m] qPochhammer[a q,q,k+m]), {k, m, n}, n, 0]

(3.47), (3.48)

qZeil[(-1)^k q^Binomial[k,2] (1-a q^(2k)) qPochhammer[a q,q,k-1] /
     (qPochhammer[q,q,k] qPochhammer[q,q,n-k] qPochhammer[a q,q,n+k]), {k, 0, n}, n, 0]

Andrews [1986b]

(4.6)

qZeil[qPochhammer[a q,q,j-1] (1-a q^(2j)) qPochhammer[b,q,j] /
     (qPochhammer[q,q,j] qPochhammer[a q/b,q,j] a^j b^j q^(j^2)) /. j->n-1-j,
      {j, 0, n-1}, n, 1]

qZeil[b^(1-n) qPochhammer[a q,q,n-1] qPochhammer[b,q,j] a^(-j) q^(j(1-n)) /
     (qPochhammer[a q/b,q,n-1] qPochhammer[q,q,j]) /. j->n-1-j,
      {j, 0, n-1}, n, 1]

(7.10), (7.11), (7.12)

qZeil[qBinomial[2n+j-1,2j-1,q] qPochhammer[q,q,j-1] (-1)^j *
      q^Binomial[2n-j,2], {j, 1, 2n}, n, 1]

qZeil[-2 q^(3n^2-2n) q^(-j^2-j) /. j->n-1-j, {j, 0, n-1}, n, 1]

qZeil[qBinomial[2n+j,2j-1,q] qPochhammer[q,q,j-1] (-1)^(1-j) *
      q^Binomial[2n+1-j,2], {j, 1, 2n+1}, n, 1]

qZeil[q^(3n^2+n) q^(-j^2) /. j->j-n, {j, 0, 2n}, n, 1]

qZeil[qBinomial[n+j-1,2j-1,q] qPochhammer[q,q,j-1] (-1)^(n-j) *
      q^Binomial[n-j,2], {j, 1, n}, n, 2]

Andrews [1992]

Lemma 6

qZeil[qBinomial[2n+1,n-k,q^2] (-1/z)^k q^(k^2+k) /
     (qPochhammer[z,q^2,n+1] qPochhammer[q^2/z,q^2,n]), {k, -n-1, n}, n, 1]

Andrews [1995]

(4.5)

qZeil[(-1)^k q^(9/2 k^2+1/2 k) qBinomial[2n,n+3k,q],
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[q^3,q^3,n-k-1] (1-q^(2n)) qPochhammer[q,q,3k] q^(3k^2) /
     (qPochhammer[q,q,n-3k] qPochhammer[q^3,q^3,2k] qPochhammer[q^3,q^3,k]),
      {k, -Infinity, Infinity}, n, 3]

(4.6)

qZeil[(-1)^k q^(9/2 k^2-5/2 k) qBinomial[2n,n+3k-1,q],
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[q^3,q^3,n-k-1] (1-q^(3k+2)-q^(n+3k+2)+q^(n+1)) *
      qPochhammer[q,q,3k] q^(3k^2+3k) /
     (qPochhammer[q,q,n-3k-1] qPochhammer[q^3,q^3,2k+1] qPochhammer[q^3,q^3,k]),
      {k, -Infinity, Infinity}, n, 3]

(4.7)

qZeil[(-1)^k q^(9/2 k^2+7/2 k) qBinomial[2n,n+3k+1,q],
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[q^3,q^3,n-k-1] (1-q^(3k+1)-q^(n+3k+2)+q^n) *
      qPochhammer[q,q,3k] q^(3k^2+3k) /
     (qPochhammer[q,q,n-3k-1] qPochhammer[q^3,q^3,2k+1] qPochhammer[q^3,q^3,k]),
      {k, -Infinity, Infinity}, n, 3]

Andrews, Bressoud [1984]

(1.2) (corrected)

qZeil[q^((s-r)(m-r)) qBinomial[s,r,q] qBinomial[m+n-s,n-s+r,q] *
      qBinomial[m+n+k+r,m+n,q] /
     (qBinomial[m+n+k,n,q] qBinomial[m+s+k,m,q]),
      {r, 0, s}, s, 1, {k, m, n}]

(5.2) (corrected)

qZeil[qBinomial[μ-s,m-r,q] qBinomial[ν+r-s,n+r-s,q] qPochhammer[q^s,q,k] *
      qPochhammer[q^(n-ν),q,k] qPochhammer[q^(m+s-μ-r),q,k] q^k /
     (qPochhammer[q,q,k] qPochhammer[q^(s-μ),q,k] qPochhammer[q^(s-r-ν),q,k]),
      {k, -Infinity, Infinity}, ν, 2, {m, μ, n, r, s}]

qZeil[q^((m+s-μ-r)(n-ν)) qBinomial[m+ν-μ,ν-n,q] *
      qBinomial[n+μ-s-ν,m+n-r-ν,q] qPochhammer[q^(m+n+1),q,k] *
      qPochhammer[q^(n-ν),q,k] qPochhammer[q^(m+s-μ-r),q,k] q^k /
     (qPochhammer[q^(m+n-μ+1),q,k] qPochhammer[q^(m+n-r-ν+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, ν, 2, {m, μ, n, r, s}]

Bailey [1947]

(3.4)

qZeil[(1+q^(3k))/2 qPochhammer[f,q,k] qPochhammer[g,q,k] qPochhammer[q^(-n),q,k] *
      (q^n/(f g))^k qPochhammer[q/f,q,n] qPochhammer[q/g,q,n] /
     (qPochhammer[q/f,q,k] qPochhammer[q/g,q,k] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,n] *
      qPochhammer[q/(f g),q,n] (1 - (1-f)(1-g)(1-q^(-n))/(1-f g q^(-n)))),
      {k, -n, n}, n, 1]

between (8.6) and (8.7)

qZeil[(1-a^2 q^(4k)) qPochhammer[a q,q,k-1] qPochhammer[e,q,k] qPochhammer[f,q,k] *
      qPochhammer[q^(-n),q,k] (a q^n/(e f))^k qPochhammer[a q/e,q,n] qPochhammer[a q/f,q,n] /
     (qPochhammer[q,q,k] qPochhammer[a q/e,q,k] qPochhammer[a q/f,q,k] qPochhammer[a q^(n+1),q,k] *
      qPochhammer[a q,q,n] qPochhammer[a q/(e f),q,n] (1+a-(1-e)(1-f)(1-q^(-n))/
      (1-e f q^(-n)/a))), {k, 0, n}, n, 1]

first identity in §10

qZeil[qPochhammer[q,q,n] qPochhammer[a q^2,q^2,k-1] (1-a q^(4k)) qPochhammer[f,q^2,k] *
      (a/f)^k q^(2k^2) qPochhammer[a q,q^2,n] qPochhammer[a q/f,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[a q^(n+1),q,2k] qPochhammer[q^2,q^2,k] *
      qPochhammer[a q^2/f,q^2,k] qPochhammer[a q,q,n] qPochhammer[a q/f,q^2,n]),
      {k, 0, n}, n, 1]

between (10.5) and (10.6) (corrected)

qZeil[qPochhammer[q,q,n] (-1)^k qPochhammer[a q^3,q^3,k-1] (1-a q^(6k)) a^k *
      q^(3k(3k-1)/2) qPochhammer[a q^(n+1),q,n-1] /
     (qPochhammer[q,q,n-3k] qPochhammer[a q^(n+1),q,3k] qPochhammer[q^3,q^3,k] *
      qPochhammer[a q^3,q^3,n-1]), {k, 0, n}, n, 1]

Bailey [1949]

§4

qZeil[qPochhammer[r1,q,k] qPochhammer[r2,q,k] qPochhammer[q^(-N),q,k] q^k qPochhammer[a q,q,N] *
      qPochhammer[a q/(r1 r2),q,N] qPochhammer[a q/r1,q,n] qPochhammer[a q/r2,q,n] *
      qPochhammer[a q^(N+1),q,n] /
     (qPochhammer[r1 r2 q^(-N)/a,q,k] qPochhammer[q,q,k-n] qPochhammer[a q,q,k+n] *
      qPochhammer[a q/r1,q,N] qPochhammer[a q/r2,q,N] (-1)^n qPochhammer[r1,q,n] *
      qPochhammer[r2,q,n] qPochhammer[q^(-N),q,n] (q x/(r1 r2))^n q^(n(2N-n+1)/2)),
      {k, n, Infinity}, N, 1, {n}]

between (5.1) and (5.2)

qZeil[qPochhammer[r1,q,k] qPochhammer[r2,q,k] (a/(r1 r2))^k qPochhammer[a q/r1,q,n] *
      qPochhammer[a q/r2,q,n] (r1 r2-a) qPochhammer[a q,q,Infinity] *
      qPochhammer[a q/(r1 r2),q,Infinity] /
     (qPochhammer[q,q,k-n] qPochhammer[a q,q,k+n] qPochhammer[r1,q,n] qPochhammer[r2,q,n] *
      (a/(r1 r2))^n (r1 r2 (1+a q^(2n)) - a q^n (r1+r2)) *
      qPochhammer[a q/r1,q,Infinity] qPochhammer[a q/r2,q,Infinity]),
      {k, 0, Infinity}, n, 1]

(5.2)

qZeil[(-1)^k qPochhammer[r1,q,k] (a/r1)^k q^Binomial[k,2] qPochhammer[a q/r1,q,n] *
      qPochhammer[a q,q,Infinity] /
     (qPochhammer[q,q,k-n] qPochhammer[a q,q,k+n] (-1)^n qPochhammer[r1,q,n] (a/r1)^n *
      q^Binomial[n,2] (1+a q^(2n)-a q^n/r1) qPochhammer[a q/r1,q,Infinity]),
      {k, 0, Infinity}, n, 1]

second identity in §6

qZeil[(-1)^k qPochhammer[p,q,k] q^Binomial[k,2] y^k /
     (qPochhammer[q,q,k] qPochhammer[q,q,n-k] qPochhammer[a q,q,n+k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[p,q,k] (y q^n)^k /
     (qPochhammer[q,q,n] qPochhammer[a q,q,n] qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

(i), p. 5

qZeil[(-1)^k (1-a q^(2k)) qPochhammer[a q,q,k-1] a^k q^(k(3k-1)/2) qPochhammer[q,q,n] /
     (qPochhammer[q,q,k] qPochhammer[q,q,n-k] qPochhammer[a q,q,n+k]),
      {k, 0, n}, n, 1]

(ii), p. 5 (corrected)

qZeil[(1-a q^(2k)) qPochhammer[a q,q,k-1] qPochhammer[b,q,k] a^k q^(k^2) *
      qPochhammer[q,q,n] qPochhammer[a q/b,q,n] /
     (qPochhammer[q,q,k] qPochhammer[a q/b,q,k] b^k qPochhammer[q,q,n-k] qPochhammer[a q,q,n+k]),
      {k, 0, n}, n, 1]

(iv), p. 6

qZeil[qPochhammer[a q^2,q^2,k-1] qPochhammer[f,q^2,k] (1-a q^(4k)) a^k q^(2k^2) *
      qPochhammer[q,q,n] qPochhammer[a q,q^2,n] qPochhammer[a q/f,q,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[a q^2/f,q^2,k] f^k qPochhammer[a q,q,n+2k] *
      qPochhammer[q,q,n-2k] qPochhammer[a q/f,q^2,n]), {k, 0, n}, n, 1]

(v), p. 6

qZeil[(-1)^k qPochhammer[a q^3,q^3,k-1] (1-a q^(6k)) a^k q^(3k(3k-1)/2) *
      qPochhammer[q,q,n] qPochhammer[a q,q,2n-1] /
     (qPochhammer[q^3,q^3,k] qPochhammer[a q,q,n+3k] qPochhammer[q,q,n-3k] qPochhammer[a q^3,q^3,n-1]),
      {k, 0, n}, n, 1]

Bressoud [1981]

(1.1)

qZeil[q^(k^2) qBinomial[n,k,q], {k, 0, n}, n, 2]

qZeil[(-1)^k q^(k(5k+1)/2) qBinomial[2n,n+2k,q],
      {k, -Infinity, Infinity}, n, 2, MagicFactor->-k]

(1.5)

qZeil[q^(n k) qBinomial[n,k,q], {k, 0, n}, n, 2]

qZeil[(-1)^k q^((3k^2+k)/2) qBinomial[2n,n+2k,q],
      {k, -Infinity, Infinity}, n, 2, MagicFactor->-k]

(1.6)

qZeil[q^(2k^2) qBinomial[n,k,q^2] qPochhammer[-q^(2n-2k+1),q^2,k] /
      qPochhammer[-q,q^2,k], {k, 0, n}, n, 2]

qZeil[(-1)^k q^(4k^2+k) qBinomial[2n,n+2k,q^2],
      {k, -Infinity, Infinity}, n, 2]

(1.7)

qZeil[q^(2k^2+2k) qBinomial[n,k,q^2] qPochhammer[-q^(2n-2k+3),q^2,k+1] /
      qPochhammer[-q,q^2,k+1], {k, 0, n}, n, 2]

qZeil[(1+q^(2n+3)) (-1)^k q^(4k^2+3k) qBinomial[2n+2,n+2k+2,q^2] /
      (1-q^(2n+2)), {k, -Infinity, Infinity}, n, 2]

(1.8) (LHS with k->n-k 1350 sec, RHS several hours)

qZeil[q^(2k^2) qBinomial[n,k,q^2] qPochhammer[-q^(2n-2k+1),q,2k] qPochhammer[q^(2n+2),q^2,k] /
     (qPochhammer[-q,q,2k] qPochhammer[q^(4n-2k+2),q^2,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 4]

qZeil[(-1)^k q^((7k^2+k)/2) qBinomial[2n,n+2k,q^2] *
      qPochhammer[q^(2n+1),q,k] qPochhammer[q^(2n+1),q,-k],
      {k, -Infinity, Infinity}, n, 4, MagicFactor->-k]

(3.4)

qZeil[a^k q^(k^2) qBinomial[n,k,q] qPochhammer[a g q,q,n] /
     (qPochhammer[a g q,q,k] qPochhammer[a g q,q,n-k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[a^2 q,q,2n] qPochhammer[q,q,n] a^(2k) g^k q^(2k^2) (1-a q^(2k)) *
      qPochhammer[a,q,k] qPochhammer[1/g,q,k] /
     (qPochhammer[a q,q,n] (1-a) qPochhammer[q,q,k] qPochhammer[a g q,q,k] qPochhammer[q,q,n-2k] *
      qPochhammer[a^2 q,q,n+2k]), {k, 0, Infinity}, n, 2]

Bressoud [1983]

Main Theorem

qZeil[(1-a q^(2k)) qPochhammer[b,q,k+n] qPochhammer[b/a,q,n-k] (b/a)^k *
      (1-b q^(2m)) qPochhammer[a,q,m+k] qPochhammer[a/b,q,k-m] (a/b)^m /
     ((1-a) qPochhammer[a q,q,k+n] qPochhammer[q,q,n-k] (1-b) qPochhammer[b q,q,m+k] *
      qPochhammer[q,q,k-m]), {k, m, n}, n, 0]

Bressoud [1987]

(1.4)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (q^(-n)/(b c))^k *
      qPochhammer[q^(-2n)/b,q,n] qPochhammer[q^(-2n)/c,q,n] /
     (qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(-2n)/c,q,k] qPochhammer[q,q,k] *
      qPochhammer[q^(-2n),q,n] qPochhammer[q^(-2n)/(b c),q,n]), {k, 0, 2n}, n, 1,
      MagicFactor->2n-k]

Carlitz [1969]

(1.1) with k->k+n, c->c q^(-n), d->d q^(-n)

qZeil[(-1)^k qBinomial[2n,k,q] qPochhammer[q^(1-3n)/(c d e),q,k] *
      qPochhammer[c,q,k] qPochhammer[d,q,k] qPochhammer[e,q,k] *
      qPochhammer[q^(1-3n)/(c d e),q,2n-k] qPochhammer[c,q,2n-k] qPochhammer[d,q,2n-k] *
      qPochhammer[e,q,2n-k] q^(k(6n-3k+1)/2) /
     ((-1)^n qPochhammer[q^(1-3n)/(c d e),q,n] qPochhammer[c,q,n] qPochhammer[d,q,n] *
      qPochhammer[e,q,n] qPochhammer[q^(n+1),q,n] qPochhammer[q^(1-2n)/(d e),q,n] *
      qPochhammer[c d q^n,q,n] qPochhammer[q^(1-2n)/(c e),q,n] e^n q^(n(3n-1)/2)) /.
      {k->k+n, c->c q^(-n), d->d q^(-n)},
      {k, -n, n}, n, 1, MagicFactor->-k]

(1.2)

qZeil[(-1)^k qBinomial[2n,k,q] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[b,q,2n-k] qPochhammer[c,q,2n-k] q^(k(2n-k+1)/2) /
     (qPochhammer[q^(n+1),q,n] qPochhammer[b c q^n,q,n] qPochhammer[b,q,n] qPochhammer[c,q,n]),
      {k, 0, 2n}, n, 1, MagicFactor->2n-k]

(1.3) with q->q^2, k->k+n, b->b q^(-2n)

qZeil[qBinomial[2n,k,q^2] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[b,q^2,2n-k] qPochhammer[c,q^2,2n-k] q^k qPochhammer[b c q,q^2,n] *
      qPochhammer[b c q,q^2,2n] qPochhammer[q,q^2,n] /
     (qPochhammer[b c q,q^2,k] qPochhammer[b c q,q^2,2n-k] qPochhammer[b,q^2,n] *
      qPochhammer[c,q^2,n] qPochhammer[q^(2n+2),q^2,n] qPochhammer[b c q^(2n),q^2,n] *
      qPochhammer[b q,q^2,n] qPochhammer[c q,q^2,n]) /. 
      {k->k+n, b->b q^(-2n)},
      {k, -n, n}, n, 1, MagicFactor->-k]

(2.5) with n=2m, b->b q^(-m), c->c q^(-m)  (corrected, LHS with k->m-k, RHS with j->m-j)

qZeil[qPochhammer[q^(-2m),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (q^(1-m) z/(b c))^k /
     (qPochhammer[q,q,k] qPochhammer[q^(1-2m)/b,q,k] qPochhammer[q^(1-2m)/c,q,k]) /.
      {b->b q^(-m), c->c q^(-m), k->m-k},
      {k, -Infinity, Infinity}, m, 3]

qZeil[(-1)^j qPochhammer[q^(-2m),q,2j] qPochhammer[q^(1-2m)/(b c),q,j] z^j *
      q^(-j(j-1)/2+m j) qPochhammer[z,q,m-j] qPochhammer[q^(j-m) z,q,m-j] /
     (qPochhammer[q,q,j] qPochhammer[q^(1-2m)/b,q,j] qPochhammer[q^(1-2m)/c,q,j]) /.
     {b->b q^(-m), c->c q^(-m), j->m-j},
      {j, -Infinity, Infinity}, m, 3]

(2.6) with n=2m (corrected)

qZeil[qPochhammer[q^(-2m),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (q^(1-m)/(b c))^k *
      qPochhammer[q,q,m] qPochhammer[q^(1-2m)/b,q,m] qPochhammer[q^(1-2m)/c,q,m] /
     (qPochhammer[q,q,k] qPochhammer[q^(1-2m)/b,q,k] qPochhammer[q^(1-2m)/c,q,k] *
      (-1)^m qPochhammer[q^(-2m),q,2m] qPochhammer[q^(1-2m)/(b c),q,m] *
      q^(m(m+1)/2)), {k, 0, 2m}, m, 1, MagicFactor->2m-k]

(2.6) with n=2m+1 (corrected)

qZeil[qPochhammer[q^(-2m-1),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (q^(-m)/(b c))^k *
      qPochhammer[q,q,m] qPochhammer[q^(-2m)/b,q,m] qPochhammer[q^(-2m)/c,q,m] /
     (qPochhammer[q,q,k] qPochhammer[q^(-2m)/b,q,k] qPochhammer[q^(-2m)/c,q,k] *
      (-1)^m qPochhammer[q^(-2m-1),q,2m] qPochhammer[q^(-2m)/(b c),q,m] *
      q^(m(m+1)/2) (1-1/q)), {k, 0, 2m+1}, m, 1, MagicFactor->2m+1-k]

(2.9) with n=2m

qZeil[(-1)^k qBinomial[2m,k,q] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[b,q,2m-k] *
      qPochhammer[c,q,2m-k] q^(k(2m-k-1)/2) /
     (qPochhammer[q^(m+1),q,m] qPochhammer[q^m b c,q,m] qPochhammer[b,q,m] qPochhammer[c,q,m] *
      q^(-m)), {k, 0, 2m}, m, 1, MagicFactor->2m-k]

(2.9) with n=2m+1

qZeil[(-1)^k qBinomial[2m+1,k,q] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[b,q,2m+1-k] qPochhammer[c,q,2m+1-k] q^(k(2m-k-1)/2) /
     (-qPochhammer[q^(m+1),q,m+1] qPochhammer[q^(m+1) b c,q,m] qPochhammer[b,q,m+1] *
      qPochhammer[c,q,m+1] q^(-2m-1)), {k, 0, 2m+1}, m, 1, MagicFactor->2m+1-k]

(2.12)

qZeil[(-1)^k qBinomial[2m+1,k,q] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[b,q,2m+1-k] qPochhammer[c,q,2m+1-k] q^(k(2m-k+3)/2) /
     (qPochhammer[q^(m+1),q,m+1] qPochhammer[q^(m+1) b c,q,m] qPochhammer[b,q,m+1] *
      qPochhammer[c,q,m+1]), {k, 0, 2m+1}, m, 1, MagicFactor->2m+1-k]

Chu [1993]

below (3.1a)

qZeil[(-1)^k qBinomial[n,k,q] q^Binomial[n-k,2] qPochhammer[a q^k,q,n] *
	  qPochhammer[a,q,k] qPochhammer[a q/(b c),q,k] qPochhammer[a q/b,q,n] qPochhammer[a q/c,q,n] /
	 (qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] (a q/(b c))^n q^Binomial[n,2] *
      qPochhammer[a,q,n] qPochhammer[b,q,n] qPochhammer[c,q,n]), {k, 0, n}, n, 1]

(5.3b) with q->q^2

qZeil[qPochhammer[b,q^2,k] qPochhammer[q^4 b,q^4,k] qPochhammer[q^2 a^2/b,q^2,k] *
      qPochhammer[b/a,q^2,k] qPochhammer[q b/a,q^2,k] qPochhammer[a^2 q^(2n),q^2,k] *
      qPochhammer[q^(-2n),q^2,k] (b q/a^2)^k qPochhammer[a q^2,q^2,n] *
      qPochhammer[a^2/b,q^2,n] qPochhammer[a,q,n] qPochhammer[-b/a,q,n] /
     (qPochhammer[b,q^4,k] qPochhammer[b^2/a^2,q^2,k] qPochhammer[q a,q,2k] *
      qPochhammer[q^(2-2n) b/a^2,q^2,k] qPochhammer[b q^(2n+2),q^2,k] qPochhammer[q^2,q^2,k] *
      q^(-n) qPochhammer[a,q^2,n] qPochhammer[b q^2,q^2,n] qPochhammer[-q,q,n] *
      qPochhammer[q a^2/b,q,n]), {k, -Infinity, Infinity}, n, 2]

CheckRec[%, 1]

(5.4b) with q->q^3

qZeil[qPochhammer[q^(-3n),q^3,k] qPochhammer[a^2 b q^(3n),q^3,k] (1-a q^(4k)) *
      qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[q/b,q,2k] q^k qPochhammer[a q^3/b,q^3,n] *
      qPochhammer[a b,q,3n] /
     (qPochhammer[q^3,q^3,k] qPochhammer[a q^3/b,q^3,k] (1-a) qPochhammer[a q^(3n+1),q,k] *
      qPochhammer[q^(-3n+1)/(a b),q,k] qPochhammer[a b,q,2k] qPochhammer[a b^2,q^3,n] *
      qPochhammer[a q,q,3n]), {k, 0, n}, n, 1]

Chu [1995a]

Example 1.1

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(-n)/(b c))^k qPochhammer[q^(-2n)/b,q,n] qPochhammer[q^(-2n)/c,q,n] /
     (qPochhammer[w,q,k] qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(-2n)/c,q,k] qPochhammer[q,q,k] *
      qPochhammer[q^(-2n),q,n] qPochhammer[q^(-2n)/(b c),q,n]),
      {k, 0, 2n}, n, 1, MagicFactor->2n-k]

Example 1.2

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(1-n)/(b c))^k (1-w) qPochhammer[q^(1-2n)/b,q,n] qPochhammer[q^(1-2n)/c,q,n] /
     (qPochhammer[w,q,k] qPochhammer[q^(1-2n)/b,q,k] qPochhammer[q^(1-2n)/c,q,k] qPochhammer[q,q,k] *
      (1-w q^n) qPochhammer[q^(-2n),q,n] qPochhammer[q^(1-2n)/(b c),q,n]),
      {k, 0, 2n}, n, 1, MagicFactor->2n-k]

Example 1.3 (corrected)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(2-n)/(b c))^k (1-b/q) (1-c/q) (1-w) qPochhammer[q^(2-2n)/b,q,n] *
      qPochhammer[q^(2-2n)/c,q,n] q^n /
     (qPochhammer[w,q,k] qPochhammer[q^(2-2n)/b,q,k] qPochhammer[q^(2-2n)/c,q,k] qPochhammer[q,q,k] *
      (1-b q^(n-1)) (1-c q^(n-1)) (1-w q^(2n)) qPochhammer[q^(-2n),q,n] *
      qPochhammer[q^(2-2n)/(b c),q,n]), {k, 0, 2n}, n, 1, MagicFactor->2n-k]

Example 1.4

qZeil[qPochhammer[q^(-1-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(-n)/(b c))^k (1-w) qPochhammer[q^(-2n)/b,q,n] qPochhammer[q^(-2n)/c,q,n] /
     (qPochhammer[w,q,k] qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(-2n)/c,q,k] qPochhammer[q,q,k] *
      (1-q^(-n-1)) qPochhammer[q^(-1-2n),q,n] qPochhammer[q^(-2n)/(b c),q,n]),
      {k, 0, 2n+1}, n, 1, MagicFactor->2n+1-k]

qZeil[qPochhammer[q^(-1-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(1-n)/(b c))^k (1-w) qPochhammer[q^(-2n)/b,q,n] qPochhammer[q^(-2n)/c,q,n] /
     (qPochhammer[w,q,k] qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(-2n)/c,q,k] qPochhammer[q,q,k] *
      q^(1+2n) w (1-q^(-n-1)) qPochhammer[q^(-1-2n),q,n] qPochhammer[q^(-2n)/(b c),q,n]),
      {k, 0, 2n+1}, n, 1, MagicFactor->2n+1-k]

Example 1.5

qZeil[qPochhammer[q^(-1-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(1-n)/(b c))^k (1-c/q) (1-w) qPochhammer[q^(-2n)/b,q,n] *
      qPochhammer[q^(1-2n)/c,q,n] /
     (qPochhammer[w,q,k] qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(1-2n)/c,q,k] qPochhammer[q,q,k] *
      (1-q^(-n-1)) (1-c w q^(2n)) qPochhammer[q^(-1-2n),q,n] *
      qPochhammer[q^(1-2n)/(b c),q,n]), {k, 0, 2n+1}, n, 1, MagicFactor->2n+1-k]

Example 1.6

qZeil[qPochhammer[q^(-1-2n),q,k] qPochhammer[q w,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      (q^(-n)/(b c))^k (1-q^(-n-1)/c) (1-w) qPochhammer[q^(-2n)/b,q,n] *
      qPochhammer[q^(-1-2n)/c,q,n] /
     (qPochhammer[w,q,k] qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(-1-2n)/c,q,k] qPochhammer[q,q,k] *
      (1-q^(-n-1)) (1-w/c) qPochhammer[q^(-1-2n),q,n] qPochhammer[q^(-2n)/(b c),q,n]),
      {k, 0, 2n+1}, n, 1, MagicFactor->2n+1-k]

Example 1.7

qZeil[qPochhammer[q^(-1-2n),q,k] qPochhammer[q u,q,k] qPochhammer[q v,q,k] qPochhammer[b,q,k] *
      qPochhammer[c,q,k] (q^(-n)/(b c))^k (1-u) (1-v) qPochhammer[q^(-2n)/b,q,n] *
      qPochhammer[q^(-2n)/c,q,n] /
     (qPochhammer[u,q,k] qPochhammer[v,q,k] qPochhammer[q^(-2n)/b,q,k] qPochhammer[q^(-2n)/c,q,k] *
      qPochhammer[q,q,k] (1-q^(-n-1)) (1-u v q^(2n+1)) qPochhammer[q^(-1-2n),q,n] *
      qPochhammer[q^(-2n)/(b c),q,n]), {k, 0, 2n+1}, n, 1, MagicFactor->2n+1-k]

Chu [1995b]

Lemma 1, first identity

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[q^m x,q,k] qPochhammer[c/b,q,m] q^k /
     (qPochhammer[q^m c,q,k] qPochhammer[x,q,k] qPochhammer[q,q,k] b^n qPochhammer[c,q,m]),
      {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c/b,q,m+n] *
      (q x/c)^k /
     (qPochhammer[x,q,k] qPochhammer[q^(1-m-n) b/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m+n]),
      {k, 0, Infinity}, n, 2, {m}]

Lemma 1, second identity

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[q^m x,q,k] qPochhammer[c,q,n] q^k /
     (qPochhammer[q^m c,q,k] qPochhammer[x,q,k] qPochhammer[q,q,k] b^n qPochhammer[c/b,q,n]),
      {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[b,q,k] qPochhammer[q^n x,q,k] qPochhammer[c,q,m] q^k /
     (qPochhammer[q^n c,q,k] qPochhammer[x,q,k] qPochhammer[q,q,k] b^m qPochhammer[c/b,q,m]),
      {k, 0, m}, n, 2]

Reversed Lemma 1, first identity (corrected)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[q^m x,q,k] qPochhammer[c/b,q,m] *
      (q^n c/b)^k / (qPochhammer[q^m c,q,k] qPochhammer[x,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m]),
      {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c/b,q,m+n] q^k /
     (qPochhammer[x,q,k] qPochhammer[q^(1-m-n) b/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m+n]),
      {k, 0, Infinity}, n, 2, {m}]

Reversed Lemma 1, second identity

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[q^m x,q,k] qPochhammer[c,q,n] *
      (q^n c/b)^k /
     (qPochhammer[q^m c,q,k] qPochhammer[x,q,k] qPochhammer[q,q,k] qPochhammer[c/b,q,n]),
      {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[b,q,k] qPochhammer[q^n x,q,k] qPochhammer[c,q,m] *
      (q^m c/b)^k /
     (qPochhammer[q^n c,q,k] qPochhammer[x,q,k] qPochhammer[q,q,k] b^m qPochhammer[c/b,q,m]),
      {k, 0, m}, n, 2]

Theorem 1 (a)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] (q^(1+m))^k qPochhammer[c/b,q,m] *
      qPochhammer[c/(a b),q,n] /
     (qPochhammer[q^m c,q,k] qPochhammer[q^(1-n) a b/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m] *
      qPochhammer[c/a,q,n]), {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] (q/a)^k *
      qPochhammer[c/b,q,m+n] /
     (qPochhammer[c/a,q,k] qPochhammer[q^(1-m-n) b/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m+n]),
      {k, 0, Infinity}, n, 2, {m}]

Theorem 1 (b)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] (q^(1+m))^k qPochhammer[c,q,n] *
      qPochhammer[c/(a b),q,n] /
     (qPochhammer[q^m c,q,k] qPochhammer[q^(1-n) a b/c,q,k] qPochhammer[q,q,k] qPochhammer[c/a,q,n] *
      qPochhammer[c/b,q,n]), {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] (q^(1+n))^k *
      qPochhammer[c,q,m] qPochhammer[c/(a b),q,m] /
     (qPochhammer[q^n c,q,k] qPochhammer[q^(1-m) a b/c,q,k] qPochhammer[q,q,k] qPochhammer[c/a,q,m] *
      qPochhammer[c/b,q,m]), {k, 0, m}, n, 2]

Reversed Theorem 1 (a)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] q^k qPochhammer[c/b,q,m] *
      qPochhammer[c/(a b),q,n] /
     (qPochhammer[q^m c,q,k] qPochhammer[q^(1-n) a b/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m] *
      qPochhammer[c/a,q,n]), {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] q^k *
      qPochhammer[c/b,q,m+n] /
     (qPochhammer[c/a,q,k] qPochhammer[q^(1-m-n) b/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,m+n]),
      {k, 0, Infinity}, n, 2, {m}]

Reversed Theorem 1 (b)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] q^k qPochhammer[c,q,n] *
      qPochhammer[c/(a b),q,n] /
     (qPochhammer[q^m c,q,k] qPochhammer[q^(1-n) a b/c,q,k] qPochhammer[q,q,k] qPochhammer[c/a,q,n] *
      qPochhammer[c/b,q,n]), {k, 0, n}, n, 2, {m}]

qZeil[qPochhammer[q^(-m),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] q^k qPochhammer[c,q,m] *
      qPochhammer[c/(a b),q,m] /
     (qPochhammer[q^n c,q,k] qPochhammer[q^(1-m) a b/c,q,k] qPochhammer[q,q,k] qPochhammer[c/a,q,m] *
      qPochhammer[c/b,q,m]), {k, 0, m}, n, 2]

Section 3, second identity with a->a^2 (corrected)

qZeil[qPochhammer[a^2,q,k] qPochhammer[-q a,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[a/b,q,n] qPochhammer[q a^2/b,q,n] qPochhammer[1/b^2,q,n] /
     (qPochhammer[-a,q,k] qPochhammer[q a^2/b,q,k] qPochhammer[b^2 q^(1-n),q,k] qPochhammer[q,q,k] *
      qPochhammer[q a/b,q,n] qPochhammer[a^2/b^2,q,n] qPochhammer[1/b,q,n]),
      {k, 0, n}, n, 1]

Section 3, third identity

qZeil[qPochhammer[a,q,k] qPochhammer[-b q^(1-n),q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[q a/b,q,n] qPochhammer[1/b^2,q,n] qPochhammer[a/b^2,q^2,n] *
      qPochhammer[q^2/b^2,q^2,n] /
     (qPochhammer[-b q^(-n),q,k] qPochhammer[q a/b,q,k] qPochhammer[b^2 q^(1-n),q,k] *
      qPochhammer[q,q,k] qPochhammer[a/b^2,q,n] qPochhammer[q/b,q,n] qPochhammer[q^2 a/b^2,q^2,n] *
      qPochhammer[1/b^2,q^2,n]), {k, 0, n}, n, 1]

Section 3, fifth identity (corrected)

qZeil[qPochhammer[a,q,k] qPochhammer[q^2 a,q^2,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[a/(b^2 q),q^2,n] qPochhammer[q a/b,q,n] qPochhammer[1/(q b^2),q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[q a/b,q,k] qPochhammer[b^2 q^(2-n),q,k] qPochhammer[q,q,k] *
      qPochhammer[q a/b^2,q^2,n] qPochhammer[a/(q b^2),q,n] qPochhammer[1/(q b),q,n]),
      {k, 0, n}, n, 1]

Section 4, second identity with a->a^2 (simplified, corrected)

qZeil[qPochhammer[a^4/b^2,q,k] qPochhammer[-q a,q,k] qPochhammer[b q^n,q,k] qPochhammer[q^(-n),q,k] *
      q^k qPochhammer[q a,q,n] qPochhammer[q b^2/a^2,q,n] qPochhammer[b/a^2,q,n] *
      qPochhammer[-a,q,n] qPochhammer[q a^2,q^2,n] /
     (qPochhammer[-a,q,k] qPochhammer[q^(1-n) a^2/b,q,k] qPochhammer[q^(1+n) a^2,q,k] *
      qPochhammer[q,q,k] qPochhammer[b/a,q,n] qPochhammer[q a^2,q,n] qPochhammer[a^2/b,q,n] *
      qPochhammer[q b^2/a^2,q^2,n] qPochhammer[-q b/a,q,n]), {k, 0, n}, n, 1]

Section 4, third identity (corrected)

qZeil[qPochhammer[a^2/b^2,q,k] qPochhammer[-q a/b,q,k] qPochhammer[b q^n,q,k] qPochhammer[q^(-n),q,k] *
      q^k qPochhammer[b/a,q,n] qPochhammer[q b^2/a,q,n] qPochhammer[q a,q^2,n] *
      qPochhammer[a q^2,q^2,n] /
     (qPochhammer[-a/b,q,k] qPochhammer[q^(1-n) a/b,q,k] qPochhammer[q^(1+n) a,q,k] *
      qPochhammer[q,q,k] qPochhammer[q a,q,n] qPochhammer[a/b,q,n] qPochhammer[b^2/a,q^2,n] *
      qPochhammer[q b^2/a,q^2,n] q^n), {k, 0, n}, n, 1]

Section 4, fourth identity with a->a^2 (corrected)

qZeil[qPochhammer[a^4/b^2,q,k] qPochhammer[q a,q,k] qPochhammer[b q^n,q,k] qPochhammer[q^(-n),q,k] *
      q^k qPochhammer[a,q,n] qPochhammer[b/a,q,n] qPochhammer[q b^2/a^2,q,n] qPochhammer[b/a^2,q,n] *
      qPochhammer[q a^2,q^2,n] qPochhammer[q^2 a^2,q^2,n] /
     (qPochhammer[a,q,k] qPochhammer[q^(1-n) a^2/b,q,k] qPochhammer[q^(1+n) a^2,q,k] *
      qPochhammer[q,q,k] qPochhammer[q a,q,n] qPochhammer[q b/a,q,n] qPochhammer[q a^2,q,n] *
      qPochhammer[a^2/b,q,n] qPochhammer[b^2/a^2,q^2,n] qPochhammer[q b^2/a^2,q^2,n]),
      {k, 0, n}, n, 1]

Section 4, fifth identity (corrected)

qZeil[qPochhammer[a^2/(q b^2),q,k] qPochhammer[q^2 a,q^2,k] qPochhammer[b q^n,q,k] *
      qPochhammer[q^(-n),q,k] q^k qPochhammer[q^2 b^2/a,q,n] qPochhammer[b/a,q,n] *
      qPochhammer[a,q^2,n] qPochhammer[q a,q^2,n] /
     (qPochhammer[a,q^2,k] qPochhammer[q^(1-n) a/b,q,k] qPochhammer[q^(1+n) a,q,k] *
      qPochhammer[q,q,k] qPochhammer[q a,q,n] qPochhammer[a/(q b),q,n] qPochhammer[q b^2/a,q^2,n] *
      qPochhammer[q^2 b^2/a,q^2,n]), {k, 0, n}, n, 1]

Section 5, second identity with a->a^2

qZeil[qPochhammer[a^2,q,k] qPochhammer[q a,q,k] qPochhammer[b q^n,q,k] qPochhammer[q^(-n),q,k] *
      q^k (1-a^2/b) (1+q^n) qPochhammer[q,q,n] qPochhammer[1/b,q,n] qPochhammer[q a^2,q^2,n] *
      qPochhammer[q^2 a^2,q^2,n] /
     (qPochhammer[a,q,k] qPochhammer[b q^(1-n),q,k] qPochhammer[q^(1+n) a^2,q,k] *
      qPochhammer[q,q,k] (1+a/b) (1-q^n a) qPochhammer[q a^2,q,n] qPochhammer[a^2/b,q,n] *
      qPochhammer[q,q^2,n] qPochhammer[q^2,q^2,n]) /. n->n+1, {k, 0, n+1}, n, 1]

Section 5, third identity (corrected)

qZeil[qPochhammer[a,q,k] qPochhammer[-q b,q,k] qPochhammer[b q^n,q,k] qPochhammer[q^(-n),q,k] *
      q^k (1+1/b) (1-a/b) qPochhammer[q,q,n] qPochhammer[1/b,q,n] qPochhammer[q a,q^2,n] *
      qPochhammer[q^2 a,q^2,n] /
     (qPochhammer[-b,q,k] qPochhammer[b q^(1-n),q,k] qPochhammer[q^(1+n) a,q,k] *
      qPochhammer[q,q,k] (1-a/b^2) qPochhammer[q a,q,n] qPochhammer[a/b,q,n] *
      qPochhammer[q,q^2,n] qPochhammer[q^2,q^2,n]) /. n->n+1, {k, 0, n+1}, n, 1]

Section 5, fourth identity with a->a^2 (simplified, corrected)

qZeil[qPochhammer[a^2,q,k] qPochhammer[q^2 a^2,q^2,k] qPochhammer[b q^n,q,k] qPochhammer[q^(-n),q,k] *
      q^k qPochhammer[q,q,n] qPochhammer[1/b,q,n] qPochhammer[q^2 a^2,q^2,n] *
      qPochhammer[q a^2,q^2,n] (1-a^2/b) (1+q^n) /
     (qPochhammer[q a,q,k] qPochhammer[-a,q,k] qPochhammer[b q^(1-n),q,k] *
      qPochhammer[q^(1+n) a^2,q,k] qPochhammer[q,q,k] qPochhammer[q a^2,q,n] qPochhammer[a^2/b,q,n] *
      qPochhammer[q,q^2,n] qPochhammer[q^2,q^2,n] (1-a/b) (1+a q^n)) /. n->n+1,
      {k, 0, n+1}, n, 1]

Cigler [1979]

(17)

qZeil[qBinomial[n+m,k+m,q] q^Binomial[k,2] x^k /
     (qPochhammer[-x,q,n] qPochhammer[-q/x,q,m]), {k, -m, n}, n, 1]

Cigler [1982]

1.2.(5')

qZeil[qBinomial[n,k,q^2] q^k / qPochhammer[-q,q,n],
      {k, 0, n}, n, 1, MagicFactor->n-k]

1.3.(2)

qZeil[(-1)^(n-k) qBinomial[n,k,q] q^(Binomial[n-k,2]) a^(n-k) x^(k-n) /
      qPochhammer[a/x,q,n], {k, 0, n}, n, 1]

1.3.(5)

qZeil[qBinomial[n,k,q] x^k qPochhammer[a/x,q,k] a^(n-k) qPochhammer[y/a,q,n-k] /
      (x^n qPochhammer[y/x,q,n]), {k, 0, n}, n, 1]

1.4.(5)

qZeil[qBinomial[n+1,k,q] a^(n+1-k) x^k, {k, 0, n+1}, n, 2]

Gasper [1989a]

(1.5) (substitution-inversion trick)

pqTelescope[(1-a q^k/p^k) (-1)^k p^Binomial[n-k,2] qPochhammer[a,q,k] /
            (qPochhammer[p,p,k] qPochhammer[p,p,n-k] qPochhammer[a/p^n,q,k+1]), {k, 0, n}]

(1.8), (1.9)

pqTelescope[(-1)^(n+j) qPochhammer[a q p^n,q,n-1] (1-a p^j q^j) *
            qPochhammer[a p^m q^m,q,j-m] p^Binomial[j-m,2] /
           (qPochhammer[p,p,n-j] qPochhammer[a q p^n,q,j] qPochhammer[p,p,j-m]), {j, m, n}]

(5.21) with f:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[d,q,k] qPochhammer[e,q,k] *
      qPochhammer[q^(-n),q,k] (a^2 q^(2+n)/(d e))^k q^(2 Binomial[k,2]) /
     (qPochhammer[q,q,k] qPochhammer[a,q^2,k] qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] *
      qPochhammer[a q^(1+n),q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[a q,q,n] qPochhammer[e,q,k] qPochhammer[q^(-n),q,k] (a q^(1+n)/e)^k /
     (qPochhammer[a q/e,q,n] qPochhammer[a q/d,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

(5.27) with e:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[d,q,k] qPochhammer[q^(-n),q,k] *
      (-a^2 q^(2+n)/d)^k q^(3 Binomial[k,2]) /
     (qPochhammer[q,q,k] qPochhammer[a,q^2,k] qPochhammer[a q/d,q,k] qPochhammer[a q^(1+n),q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[a q,q,n] qPochhammer[q^(-n),q,k] (a q^(1+n))^k q^Binomial[k,2] /
     (qPochhammer[a q/d,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

Gasper [1989b]

(2.11), (2.12) with E^(I θ)->e

qZeil[qPochhammer[b,q,2n] e^(2n) qPochhammer[q^(-2n),q,k] qPochhammer[b,q,k] (q/(b e^2))^k /
     (qPochhammer[q,q,2n] qPochhammer[q^(1-2n)/b,q,k] qPochhammer[q,q,k]),
      {k, 0, 2n}, n, 2]

qZeil[qPochhammer[b^2,q,2n] qPochhammer[-q,q,n] qPochhammer[-q^(1/2),q,n] q^(-n/2) *
      qPochhammer[q^(-n),q,k] qPochhammer[b q^n,q,k] qPochhammer[q^(1/2) e^2,q,k] *
      qPochhammer[q^(1/2)/e^2,q,k] q^k /
     (qPochhammer[q,q,2n] qPochhammer[-b,q,n] qPochhammer[-b q^(1/2),q,n] *
      qPochhammer[b q^(1/2),q,k] qPochhammer[-q^(1/2),q,k] qPochhammer[q^2,q^2,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[b^2,q,2n] qPochhammer[q^(-n),q,k] qPochhammer[b q^n,q,k] *
      qPochhammer[b e^2,q,k] qPochhammer[b/e^2,q,k] q^k /
     (qPochhammer[q,q,2n] b^n qPochhammer[b^2 q,q^2,k] qPochhammer[-b,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

(2.22) with a^2->a,  b^2->b

qZeil[qPochhammer[q^(-2n),q^2,k] (1+q^(-n+2k)) qPochhammer[a,q^2,k] *
      qPochhammer[b,q^2,k] qPochhammer[q^(1-2n)/(a b),q^2,k] q^(2k) *
      qPochhammer[a b,q,n] qPochhammer[a,q^2,n] qPochhammer[b,q^2,n] /
     ((1+q^(-n)) qPochhammer[q^(2-2n)/a,q^2,k] qPochhammer[q^(2-2n)/b,q^2,k] *
      qPochhammer[a b q,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[-1,q,n] *
      qPochhammer[a,q,n] qPochhammer[b,q,n] qPochhammer[a b,q^2,n]), {k, 0, n}, n, 1]

Gasper [1997]

(3."0")

qZeil[qPochhammer[q^(j-n),q,k] qPochhammer[c/a,q,k] q^k qPochhammer[a,q,n] qPochhammer[c,q,j] /
     (qPochhammer[q^(1-n)/a,q,k] qPochhammer[q,q,k] qPochhammer[c,q,n] qPochhammer[a,q,j]),
      {k, 0, Infinity}, n, 1, {j}]

(3.1) (RHS with k->n-k)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] q^k / (qPochhammer[c,q,k] *
      qPochhammer[d,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c/a,q,k] qPochhammer[q^(1-n)/d,q,k] q^k *
      qPochhammer[a,q,n] qPochhammer[d/b,q,n] b^n /
     (qPochhammer[q^(1-n)/a,q,k] qPochhammer[q^(1-n) b/d,q,k] qPochhammer[q,q,k] *
      qPochhammer[c,q,n] qPochhammer[d,q,n]) /. k->n-k, {k, 0, n}, n, 2]

(5.7)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[a/b,q,k] qPochhammer[b,q,k] *
      qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[a q^(1+n)/e,q,k] (q^(1-n)/d)^k *
      qPochhammer[a q/e,q,n] qPochhammer[q/e,q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] *
      qPochhammer[b q,q,k] qPochhammer[e q^(-n),q,k] qPochhammer[q,q,k] qPochhammer[a q/(b e),q,n] *
      qPochhammer[b q/e,q,n]), {k, 0, Infinity}, n, 2]

CheckRec[%, 1]

(5.12) (corrected)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[a/b,q,k] qPochhammer[b,q,k] *
      qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[a q^(1+n)/e,q,k] *
      qPochhammer[a q^(1+m)/f,q,k] (q^(1-m-n)/d)^k qPochhammer[a q/e,q,n] *
      qPochhammer[q/e,q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] *
      qPochhammer[a q/f,q,k] qPochhammer[b q,q,k] qPochhammer[f q^(-m),q,k] *
      qPochhammer[e q^(-n),q,k] qPochhammer[q,q,k] qPochhammer[a q/(b e),q,n] *
      qPochhammer[b q/e,q,n]), {k, -Infinity, Infinity}, n, 3, {m}]

CheckRec[%, 1]

Gasper, Rahman [1990a]

(3.7)

qZeil[(1-a c q^(4k)) qPochhammer[a,q,k] qPochhammer[q/a,q,k] qPochhammer[a c,q,2k] *
      qPochhammer[a c q^(3n+1),q^3,k] qPochhammer[q^(-3n),q^3,k] q^k *
      qPochhammer[q,q^3,n] qPochhammer[q^2,q^3,n] qPochhammer[a^2 c q^2,q^3,n] *
      qPochhammer[c q^3,q^3,n] /
     ((1-a c) qPochhammer[c q^3,q^3,k] qPochhammer[a^2 c q^2,q^3,k] *
      qPochhammer[q,q,2k] qPochhammer[q^(-3n),q,k] qPochhammer[a c q^(3n+1),q,k] *
      qPochhammer[a q,q^3,n] qPochhammer[q^2/a,q^3,n] qPochhammer[a c q^2,q^3,n] *
      qPochhammer[a c q^3,q^3,n]), {k, 0, n}, n, 1]

(4.10)

qZeil[qPochhammer[a/b,q^3,k] qPochhammer[a q^6/b,q^6,k] qPochhammer[1/(a b^2),q^3,k] *
      qPochhammer[q^(-3n),q^3,k] qPochhammer[a q^n,q,3k] q^(3k) *
      qPochhammer[a^2 b q^3,q^3,n] qPochhammer[b,q,n] qPochhammer[q/b,q,2n] /
     (qPochhammer[a/b,q^6,k] qPochhammer[a^2 b q^3,q^3,k] qPochhammer[a/b q^(3+3n),q^3,k] *
      qPochhammer[q^(1-n)/b,q,3k] qPochhammer[q^3,q^3,k] qPochhammer[a q^3/b,q^3,n] *
      qPochhammer[1/(a b),q,n] qPochhammer[a b q,q,2n]), {k, 0, n}, n, 1]

Gasper, Rahman [1990b]

(II.1) with z:=q

qZeil[q^k / qPochhammer[q,q,k], {k, 0, n}, n, 0]

(II.1) with z:=q^(n+1)

qZeil[q^((n+1) k) qPochhammer[q^(n+1),q,Infinity] / qPochhammer[q,q,k],
      {k, 0, Infinity}, n, 1]

(II.2) with z:=q^n

qZeil[q^Binomial[k,2] q^(n k) / (qPochhammer[q,q,k] qPochhammer[-q^n,q,Infinity]),
      {k, 0, Infinity}, n, 1]

(II.3) with z:=q

qZeil[qPochhammer[a,q,k] q^k / qPochhammer[q,q,k], {k, 0, n}, n, 0]

(II.4)

qZeil[qPochhammer[q^(-n),q,k] z^k / (qPochhammer[q,q,k] qPochhammer[z q^(-n),q,n]),
      {k, 0, n}, n, 1]

(II.5) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q,n] q^(k(k-1)/2) (-c q^n)^k /
     (qPochhammer[c,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 1]

(II.6)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[c,q,n] q^k /
     (qPochhammer[c,q,k] qPochhammer[q,q,k] qPochhammer[c/a,q,n] a^n), {k, 0, n}, n, 1]

(II.7)

qZeil[qPochhammer[a,q,k] qPochhammer[q^(-n),q,k] qPochhammer[c,q,n] (c/a q^n)^k /
     (qPochhammer[c,q,k] qPochhammer[q,q,k] qPochhammer[c/a,q,n]), {k, 0, n}, n, 1]

(II.9) with b:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[q^(-n),q,k] (-q^(n+1))^k qPochhammer[a q^(1+n),q,n] /
     (qPochhammer[a q^(1+n),q,k] qPochhammer[q,q,k] qPochhammer[-q,q,n] qPochhammer[a q,q^2,n]),
      {k, 0, n}, n, 1]

(II.10) with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q^(2n+1),q,k] b^k q^(Binomial[k,2]) *
      qPochhammer[b,q,2n] /
     (qPochhammer[-q,q,k] qPochhammer[b,q,k] qPochhammer[q,q,k] qPochhammer[q^(-2n) b,q^2,2n]),
      {k, 0, 2n}, n, 1]

(II.11) with a:=q^(-n)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[b^2,q,k] q^(k + Binomial[k,2]) *
      qPochhammer[b^2 q^(1-2n),q^2,n] /
     (qPochhammer[b^2 q^(1-2n),q^2,k] qPochhammer[q,q,k] qPochhammer[q^(1-2n),q^2,n]),
      {k, 0, 2n}, n, 1]

(II.12)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,n] *
      qPochhammer[c/(a b),q,n] q^k /
     (qPochhammer[q,q,k] qPochhammer[c,q,k] qPochhammer[a b/c q^(-n+1),q,k] *
      qPochhammer[c/a,q,n] qPochhammer[c/b,q,n]), {k, 0, n}, n, 1]

(II.14) with a->a^2

qZeil[qPochhammer[a^2,q,k] qPochhammer[-a q,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] *
      qPochhammer[a q,q,n] qPochhammer[a^2/b q,q,n] (a/b q^(1+n))^k /
     (qPochhammer[-a,q,k] qPochhammer[a^2/b q,q,k] qPochhammer[a^2 q^(n+1),q,k] *
      qPochhammer[q,q,k] qPochhammer[a^2 q,q,n] qPochhammer[a/b q,q,n]),
      {k, 0, n}, n, 1]

(II.15)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (q^(2-n)/(b c))^k *
      qPochhammer[q,q,n] qPochhammer[b c,q,n] qPochhammer[b,q,2n] qPochhammer[c,q,2n] /
     (qPochhammer[q^(1-2n)/b,q,k] qPochhammer[q^(1-2n)/c,q,k] qPochhammer[q,q,k] *
      qPochhammer[b,q,n] qPochhammer[c,q,n] qPochhammer[q,q,2n] qPochhammer[b c,q,2n]),
      {k, 0, 2n}, n, 1, MagicFactor->2n-k]

(II.16) with a:=q^(-2n)

qZeil[qPochhammer[λ,q,k] qPochhammer[λ q^2,q^2,k] qPochhammer[q^(-2n),q,k] *
      qPochhammer[λ^2 q^(1+2n)/c^2,q,k] qPochhammer[c^2,q^2,k] *
      qPochhammer[λ q/c^2,q,k] (-c^2/λ)^k qPochhammer[λ^2 q^2/c^2,q^2,n] *
      qPochhammer[c^2 q,q^2,n] qPochhammer[c^2 q^(-2n)/λ,q,2n] /
     (qPochhammer[λ,q^2,k] qPochhammer[λ q^(1+2n),q,k] qPochhammer[c^2/λ q^(-2n),q,k] *
      qPochhammer[λ^2 q^2/c^2,q^2,k] qPochhammer[c^2,q,k] qPochhammer[q,q,k] *
      qPochhammer[λ q,q,2n] qPochhammer[q^(1-2n),q^2,n] *
      qPochhammer[c^4/λ^2 q^(-2n),q^2,n]), {k, 0, 2n}, n, 1]

(II.17) with n->2n

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[a q^(2n),q,k] qPochhammer[c^2,q^2,k] q^k *
      qPochhammer[a q,q^2,n] qPochhammer[c^2 q,q^2,n] /
     (qPochhammer[a q,q^2,k] qPochhammer[c^2,q,k] qPochhammer[q,q,k] c^(2n) *
      qPochhammer[q,q^2,n] qPochhammer[a/c^2 q,q^2,n]), {k, 0, 2n}, n, 1]

(II.18) with a:=q^(-2n)

qZeil[qPochhammer[-c,q,k] qPochhammer[-c q^2,q^2,k] qPochhammer[q^(-2n),q,k] *
      qPochhammer[q^(1+2n),q,k] qPochhammer[c,q,k] qPochhammer[-d,q,k] qPochhammer[-q/d,q,k] c^k *
      qPochhammer[-c q^(-2n),q,2n] qPochhammer[c d,q,2n] qPochhammer[c q/d,q,2n] /
     (qPochhammer[-c,q^2,k] qPochhammer[-c q^(1+2n),q,k] qPochhammer[-c q^(-2n),q,k] *
      qPochhammer[c q/d,q,k] qPochhammer[c d,q,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[-c q,q,2n] qPochhammer[c d q^(-2n),q^2,2n] *
      qPochhammer[c/d q^(1-2n),q^2,2n]), {k, 0, 2n}, n, 1]

(II.19) with n->2n (corrected)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q^(2n+1),q,k] qPochhammer[c^2,q^2,k] q^k *
      qPochhammer[e,q,2n] qPochhammer[c^2 q/e,q,2n] q^(-n(2n+1)) /
     (qPochhammer[e,q,k] qPochhammer[c^2/e q,q,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[e q^(-2n),q^2,2n] qPochhammer[c^2 q^(1-2n)/e,q^2,2n]),
      {k, 0, 2n}, n, 1]

(II.21)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[q^(-n),q,k] qPochhammer[a/b q,q,n] qPochhammer[a/c q,q,n] *
      (a q^(n+1)/(b c))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a q^(n+1),q,k] *
      qPochhammer[q,q,k] qPochhammer[a q,q,n] qPochhammer[a/(b c) q,q,n]),
      {k, 0, n}, n, 1]

(II.22)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[d,q,k] qPochhammer[a^2 q^(n+1)/(b c d),q,k] qPochhammer[q^(-n),q,k] *
      qPochhammer[a/b q,q,n] qPochhammer[a/c q,q,n] qPochhammer[a/d q,q,n] *
      qPochhammer[a/(b c d) q,q,n] q^k /
     (qPochhammer[a,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[b c d/a q^(-n),q,k] qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k] *
      qPochhammer[a q,q,n] qPochhammer[a/(b c) q,q,n] qPochhammer[a/(b d) q,q,n] *
      qPochhammer[a/(c d) q,q,n]), {k, 0, n}, n, 1]

(II.27) with a:=q^(-n), r:=4, mi:=1

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b1 q,q,k] qPochhammer[b2 q,q,k] *
      qPochhammer[b3 q,q,k] qPochhammer[b4 q,q,k] q^((n-4) k) /
     (qPochhammer[b1,q,k] qPochhammer[b2,q,k] qPochhammer[b3,q,k] qPochhammer[b4,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 0]

(II.28) with z:=q^n

qZeil[q^(k^2 + 2n k) / (qPochhammer[q^2,q^2,Infinity] *
      qPochhammer[-q^(2n+1),q^2,Infinity] qPochhammer[-q^(-2n+1),q^2,Infinity]),
      {k, -Infinity, Infinity}, n, 1]

(II.29) with a:=q^(-n), b:=q^(n+1)

qZeil[qPochhammer[q^(-n),q,k] z^k qPochhammer[q^(n+1),q,n] /
     (qPochhammer[q^(n+1),q,k] qPochhammer[q,q,n] qPochhammer[z q^(-n),q,n] *
      qPochhammer[q^(1+n)/z,q,n]), {k, -n, n}, n, 1]

(II.30) with b:=q^(-n), c:=a q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a q^(-n),q,k] (-q^(2n+1))^k *
      qPochhammer[q^(1+n),q,n] qPochhammer[a q^(1+n),q,Infinity] *
      qPochhammer[q^(1+n),q,Infinity] qPochhammer[q^(1+n)/a,q,Infinity] *
      qPochhammer[-q^(2n+1),q,Infinity] /
     (qPochhammer[a q^(n+1),q,k] qPochhammer[q^(n+1),q,k] *
      qPochhammer[a q^(2+2n),q^2,Infinity] qPochhammer[q^(2+2n)/a,q^2,Infinity] *
      qPochhammer[q^2,q^2,Infinity] qPochhammer[a q,q^2,Infinity] *
      qPochhammer[q/a,q^2,Infinity]), {k, -n, n}, n, 1]

(II.31) with b:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] (q^(1+n)/(c d))^k *
      qPochhammer[q/c,q,n] qPochhammer[q/d,q,n] /
     (qPochhammer[q^(1+n),q,k] qPochhammer[q/c,q,k] qPochhammer[q/d,q,k] qPochhammer[q,q,n] *
      qPochhammer[q/(c d),q,n]), {k, -n, n}, n, 1, MagicFactor->-k]

(II.32) with b:=q^(-n), c:=a q^(-n)

qZeil[qPochhammer[a q^2,q^2,k] qPochhammer[q^(-n),q,k] qPochhammer[a q^(-n),q,k] *
      qPochhammer[d,q,k] qPochhammer[e,q,k] (a q^(1+2n)/(d e))^k *
      qPochhammer[q^(1+n),q,n] qPochhammer[a q/d,q,n] qPochhammer[a q/e,q,n] *
      qPochhammer[q/d,q,n] qPochhammer[q/e,q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[a q^(n+1),q,k] qPochhammer[q^(n+1),q,k] *
      qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] qPochhammer[a q,q,n] qPochhammer[q,q,n] *
      qPochhammer[q/a,q,n] qPochhammer[a q/(d e),q,2n]), {k, -n, n}, n, 1]

(II.33) with b:=q^(-n), c:=a q^(-n)

qZeil[qPochhammer[-Sqrt[a] q,q,k] qPochhammer[q^(-n),q,k] qPochhammer[a q^(-n),q,k] *
      qPochhammer[d,q,k] (Sqrt[a] q^(1+2n)/d)^k qPochhammer[q^(1+n),q,n] *
      qPochhammer[a q/d,q,n] qPochhammer[q/d,q,n] qPochhammer[Sqrt[a] q,q,n] *
      qPochhammer[q/Sqrt[a],q,n] /
     (qPochhammer[-Sqrt[a],q,k] qPochhammer[a q^(1+n),q,k] qPochhammer[q^(1+n),q,k] *
      qPochhammer[a q/d,q,k] qPochhammer[a q,q,n] qPochhammer[q,q,n] qPochhammer[q/a,q,n] *
      qPochhammer[Sqrt[a] q/d,q,2n]), {k, -n, n}, n, 1]

(II.34)

pqTelescope[(1-a p^k q^k)/(1-a) qPochhammer[a,p,k] qPochhammer[c,q,k] /
           (qPochhammer[q,q,k] qPochhammer[a p/c,p,k] c^k), {k, 0, n}]

(II.35)

pqTelescope[(1-a p^k q^k)(1-b p^k q^(-k)) qPochhammer[a,p,k] qPochhammer[b,p,k] *
            qPochhammer[c,q,k] qPochhammer[a/(b c),q,k] q^k /
           ((1-a)(1-b) qPochhammer[q,q,k] qPochhammer[a q/b,q,k] qPochhammer[a p/c,p,k] *
            qPochhammer[b c p,p,k]), {k, 0, n}]

(II.36)

pqTelescope[(1-a d p^k q^k)(1-b/d p^k q^(-k)) qPochhammer[a,p,k] *
            qPochhammer[b,p,k] qPochhammer[c,q,k] qPochhammer[a d^2/(b c),q,k] q^k /
           ((1-a d)(1-b/d) qPochhammer[d q,q,k] qPochhammer[a d/b q,q,k] *
            qPochhammer[a d p/c,p,k] qPochhammer[b c p/d,p,k]), {k, -m, n}]

(II.37) (corrected, substitution-inversion trick) (cf. (3.6.18))

pqTelescope[(-1)^k (1-b/a) (1-a q^k/p^k) (1-b p^k q^k) qPochhammer[a,q,k] *
            qPochhammer[b,q,k] qPochhammer[b/a p^(k+1),p,n-1] p^Binomial[n-k,2] /
           (qPochhammer[p,p,k] qPochhammer[p,p,n-k] qPochhammer[a p^(-n),q,k+1] *
            qPochhammer[b p^n,q,k+1]), {k, 0, n}]

(III.4), (III.5) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] z^k / (qPochhammer[c,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[z q^(-n),q,n] qPochhammer[q^(-n),q,k] qPochhammer[c/b,q,k] (-b z)^k *
      q^Binomial[k,2] /
     (qPochhammer[c,q,k] qPochhammer[z q^(-n),q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[b z/c q^(-n),q,n] qPochhammer[q^(-n),q,k] qPochhammer[c/b,q,k] q^k /
     (qPochhammer[c,q,k] qPochhammer[c/(b z) q,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

(III.6), (III.7), (III.8) (corrected)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] z^k / (qPochhammer[c,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[q/z,q,k] qPochhammer[q^(1-n)/c,q,k] qPochhammer[c/b,q,n] /
     (qPochhammer[b/c q^(1-n),q,k] qPochhammer[q,q,k] qPochhammer[c,q,n]) *
      (b z)^n q^(k-n), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[b z/c q^(-n),q,k] *
      qPochhammer[c/b,q,n] q^k /
     (qPochhammer[b q^(1-n)/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,n]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[q/z,q,k] qPochhammer[c/b,q,n] *
      (-z/c)^k q^(-Binomial[k,2]) b^n /
     (qPochhammer[b q^(1-n)/c,q,k] qPochhammer[q,q,k] qPochhammer[c,q,n]), {k, 0, n}, n, 2]

(III.11), (III.12)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] q^k /
     (qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[d/b,q,k] qPochhammer[d/c,q,k] *
      qPochhammer[d e/(b c),q,n] (b c/d)^n q^k /
     (qPochhammer[d,q,k] qPochhammer[d e/(b c),q,k] qPochhammer[q,q,k] qPochhammer[e,q,n]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] qPochhammer[d/b,q,k] *
      qPochhammer[e/c,q,n] c^n (b q/e)^k /
     (qPochhammer[d,q,k] qPochhammer[c/e q^(1-n),q,k] qPochhammer[q,q,k] qPochhammer[e,q,n]),
      {k, 0, n}, n, 2]

(III.13)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (d e q^n/(b c))^k /
     (qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] qPochhammer[d/b,q,k] qPochhammer[e/c,q,n] q^k /
     (qPochhammer[d,q,k] qPochhammer[c/e q^(1-n),q,k] qPochhammer[q,q,k] qPochhammer[e,q,n]),
      {k, 0, n}, n, 2]

(III.14) (RHS with k->n-k 210 sec)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] (q^(1-n) z/(b c))^k /
     (qPochhammer[q^(1-n)/b,q,k] qPochhammer[q^(1-n)/c,q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[z q^(-n),q,n] qPochhammer[q^(-n),q,2k] qPochhammer[q^(1-n)/(b c),q,k] q^k /
     (qPochhammer[q^(1-n)/b,q,k] qPochhammer[q^(1-n)/c,q,k] qPochhammer[z q^(-n),q,k] *
      qPochhammer[q/z,q,k] qPochhammer[q,q,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 3]

(III.15), (III.16)

With[{f = a b c q^(1-n) / (d e)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] q^k /
     (qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]]

With[{f = a b c q^(1-n) / (d e)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[d/b,q,k] qPochhammer[d/c,q,k] *
      qPochhammer[e/a,q,n] qPochhammer[f/a,q,n] a^n q^k /
     (qPochhammer[d,q,k] qPochhammer[a/e q^(1-n),q,k] qPochhammer[q,q,k] *
      qPochhammer[a/f q^(1-n),q,k] qPochhammer[e,q,n] qPochhammer[f,q,n]),
      {k, 0, n}, n, 2]]

With[{f = a b c q^(1-n) / (d e)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[e/a,q,k] qPochhammer[d/a,q,k] *
      qPochhammer[e d/(a b c),q,k] qPochhammer[a,q,n] qPochhammer[e d/(a b),q,n] *
      qPochhammer[e d/(a c),q,n] q^k /
     (qPochhammer[e d/(a b),q,k] qPochhammer[e d/(a c),q,k] qPochhammer[q^(1-n)/a,q,k] *
      qPochhammer[q,q,k] qPochhammer[e,q,n] qPochhammer[d,q,n] qPochhammer[e d/(a b c),q,n]),
      {k, 0, n}, n, 2]]

(III.18)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[q^(-n),q,k] (a^2 q^(2+n)/(b c d e))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[a/e q,q,k] qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[a/(b c) q,q,k] qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[q^(-n),q,k] *
      qPochhammer[a q,q,n] qPochhammer[a/(d e) q,q,n] q^k /
     (qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[d e/a q^(-n),q,k] *
      qPochhammer[q,q,k] qPochhammer[a/d q,q,n] qPochhammer[a/e q,q,n]),
      {k, 0, n}, n, 2]

(III.19)

With[{d = a b c q^(1-n) / (e f), σ = e f / (a q)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] q^k /
     (qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]]

With[{d = a b c q^(1-n) / (e f), σ = e f / (a q)},
qZeil[qPochhammer[σ,q,k] qPochhammer[σ q^2,q^2,k] qPochhammer[f/a,q,k] qPochhammer[e/a,q,k] *
      qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[q^(-n),q,k] (e f q^n/(b c))^k *
      qPochhammer[d/b,q,n] qPochhammer[d/c,q,n] /
     (qPochhammer[σ,q^2,k] qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[e f/(a b),q,k] *
      qPochhammer[e f/(a c),q,k] qPochhammer[e f/a q^n,q,k] qPochhammer[q,q,k] *
      qPochhammer[d,q,n] qPochhammer[d/(b c),q,n]), {k, 0, n}, n, 2]]

(III.20)

With[d = a b c q^(1-n) / (e f), μ = a b c / f},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] q^k /
     (qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]]

With[d = a b c q^(1-n) / (e f), μ = a b c / f},
qZeil[qPochhammer[μ,q,k] qPochhammer[μ q^2,q^2,k] qPochhammer[a,q,k] qPochhammer[b,q,k] *
      qPochhammer[c,q,k] qPochhammer[d q^n,q,k] qPochhammer[e q^n,q,k] (d e/(a b c))^k /
     (qPochhammer[μ,q^2,k] qPochhammer[μ/a q,q,k] qPochhammer[μ/b q,q,k] qPochhammer[μ/c q,q,k] *
      qPochhammer[e,q,k] qPochhammer[d,q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]]

(III.21) with c:=q^(-n)

qZeil[qPochhammer[a^2,q,k] qPochhammer[b^2,q,k] qPochhammer[q^(-n),q,k] qPochhammer[d,q,k] q^k /
     (qPochhammer[a^2 b^2 q,q^2,k] qPochhammer[-d q^(-n),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[b^2,q^2,k] qPochhammer[q^(-2n),q^2,k] *
      qPochhammer[d^2,q^2,k] q^(2k) /
     (qPochhammer[a^2 b^2 q,q^2,k] qPochhammer[-d q^(-n),q,2k] qPochhammer[q^2,q^2,k]),
      {k, 0, n}, n, 2]

(III.23) with f:=q^(-n)

With[{c = a^2 q / (b λ d)},
qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[q^(-n),q,k] *
      (a^2 q^(2+n)/(b c d e))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] qPochhammer[a q/d,q,k] *
      qPochhammer[a q/e,q,k] qPochhammer[a q^(1+n),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]]

With[{c = a^2 q / (b λ d)},
qZeil[qPochhammer[λ,q,k] qPochhammer[λ q^2,q^2,k] qPochhammer[λ b/a,q,k] *
      qPochhammer[λ c/a,q,k] qPochhammer[λ d/a,q,k] qPochhammer[e,q,k] *
      qPochhammer[q^(-n),q,k] (a q^(1+n)/e)^k  qPochhammer[a q,q,n] *
      qPochhammer[λ q/e,q,n] /
     (qPochhammer[λ,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] *
      qPochhammer[a q/d,q,k] qPochhammer[λ q/e,q,k] qPochhammer[λ q^(1+n),q,k] *
      qPochhammer[q,q,k] qPochhammer[a q/e,q,n] qPochhammer[λ q,q,n]),
      {k, 0, n}, n, 2]]

(III.25)

With[{c = a^2 q / (b d λ)},
qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] qPochhammer[q^(-n),q,k] *
      q^k /
     (qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[a^2/λ^2 q^(-n),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]]

With[{c = a^2 q / (b d λ)},
qZeil[qPochhammer[λ q/a,q,n] qPochhammer[λ^2 q/a,q,n] qPochhammer[λ,q,k] *
      qPochhammer[λ q^2,q^2,k] qPochhammer[b λ/a,q,k] qPochhammer[c λ/a,q,k] *
      qPochhammer[d λ/a,q,k] qPochhammer[a,q,2k] qPochhammer[λ^2/a q^(n+1),q,k] *
      qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[λ q,q,n] qPochhammer[λ^2 q/a^2,q,n] qPochhammer[λ,q^2,k] *
      qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[λ^2/a q,q,2k] qPochhammer[a/λ q^(-n),q,k] *
      qPochhammer[λ q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]]

(III.26) with b->b q^(-n), c->c q^(-n) (out of memory)

With[{μ = q^(1-2n) / (b c d)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] qPochhammer[e,q,k] *
      q^k /
     (qPochhammer[q^(1-n)/b,q,k] qPochhammer[q^(1-n)/c,q,k] qPochhammer[q^(1-n)/d,q,k] *
      qPochhammer[e q^(-2n)/μ^2,q,k] qPochhammer[q,q,k]) /. {b->b q^(-n), c->c q^(-n)},
      {k, -Infinity, Infinity}, n, 3]]

With[{μ = q^(1-2n) / (b c d)},
qZeil[qPochhammer[μ^2 q^(n+1),q,n] qPochhammer[μ q/e,q,n] qPochhammer[μ,q,k] *
      (1-μ q^(2k)) qPochhammer[μ b q^n,q,k] qPochhammer[μ c q^n,q,k] *
      qPochhammer[μ d q^n,q,k] qPochhammer[q^(-n),q,2k] qPochhammer[e,q,k] *
      qPochhammer[μ^2 q^(n+1)/e,q,k] q^k /
     (qPochhammer[μ^2 q^(n+1)/e,q,n] qPochhammer[μ q,q,n] (1-μ) *
      qPochhammer[q^(1-n)/b,q,k] qPochhammer[q^(1-n)/c,q,k] qPochhammer[q^(1-n)/d,q,k] *
      qPochhammer[μ^2 q^(n+1),q,2k] qPochhammer[μ q/e,q,k] qPochhammer[e q^(-n)/μ,q,k] *
      qPochhammer[q,q,k]) /. {b->b q^(-n), c->c q^(-n)}, {k, -Infinity, Infinity}, n, 3]]

(III.27)

With[{c = a^2 q / (b d λ)},
qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] *
      qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[a^2/λ^2 q^(2-n),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]]

With[{c = a^2 q / (b d λ)},
qZeil[qPochhammer[λ,q,k] qPochhammer[λ q^2,q^2,k] qPochhammer[b λ/a,q,k] *
      qPochhammer[c λ/a,q,k] qPochhammer[d λ/a,q,k] qPochhammer[a q,q,2k] *
      qPochhammer[λ^2/a q^(n-1),q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[λ/(a q),q,n] qPochhammer[λ^2/(a q),q,n] (1-λ^2/a q^(2n-1)) /
     (qPochhammer[λ,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[λ^2/a,q,2k] qPochhammer[a q^(2-n)/λ,q,k] *
      qPochhammer[λ q^(n+1),q,k] qPochhammer[q,q,k] qPochhammer[λ q,q,n] *
      qPochhammer[λ^2/(a^2 q),q,n] (1-λ^2/(a q))),
      {k, -Infinity, Infinity}, n, 3]]

(III.28)

With[{c = a^2 q / (b d λ)},
qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[λ a/(e f) q^(n+1),q,k] *
      qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[a/e q,q,k] qPochhammer[a/f q,q,k] qPochhammer[e f/λ q^(-n),q,k] *
      qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]]

With[{c = a^2 q / (b d λ)},
qZeil[qPochhammer[λ,q,k] qPochhammer[λ q^2,q^2,k] qPochhammer[b λ/a,q,k] *
      qPochhammer[c λ/a,q,k] qPochhammer[d λ/a,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] *
      qPochhammer[λ a/(e f) q^(n+1),q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[a q,q,n] qPochhammer[a/(e f) q,q,n] qPochhammer[λ/e q,q,n] *
      qPochhammer[λ/f q,q,n] /
     (qPochhammer[λ,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a/c q,q,k] qPochhammer[a/d q,q,k] *
      qPochhammer[λ/e q,q,k] qPochhammer[λ/f q,q,k] qPochhammer[e f/a q^(-n),q,k] *
      qPochhammer[λ q^(n+1),q,k] qPochhammer[q,q,k] qPochhammer[a/e q,q,n] *
      qPochhammer[a/f q,q,n] qPochhammer[λ/(e f) q,q,n] qPochhammer[λ q,q,n]),
      {k, -Infinity, Infinity}, n, 2]]

(1.6.1) with z:=q^n, q->q^2

qZeil[(-1)^k q^(k^2+2n k) / (qPochhammer[q^(1+2n),q^2,Infinity] *
      qPochhammer[q^(1-2n),q^2,Infinity] qPochhammer[q^2,q^2,Infinity]),
      {k, -Infinity, Infinity}, n, 1]

(1.6.2) with z:=q^n, q->q^2

qZeil[(-1)^k q^(k^2+2n k) / (qPochhammer[q^2,q^2,k] *
      qPochhammer[q^(1+2n),q^2,Infinity]), {k, 0, Infinity}, n, 1]

(1.6.3) with z:=q^n (corrected)

qZeil[q^(k^2+n k) qPochhammer[q^(1+n),q,Infinity] /
     (qPochhammer[q,q,k] qPochhammer[q^(1+n),q,k]), {k, 0, Infinity}, n, 1]

between (1.6.4) and (1.6.5) (corrected)

qZeil[q^(k^2+n k) qPochhammer[q,q,Infinity] /
     (qPochhammer[q,q,n] qPochhammer[q,q,k] qPochhammer[q^(n+1),q,k]),
      {k, 0, Infinity}, n, 1]

(1.9.5)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[b1 q^m,q,k] (q^(1-m)/a)^k qPochhammer[b1,q,m] /
     (qPochhammer[b q,q,k] qPochhammer[b1,q,k] qPochhammer[q,q,k] qPochhammer[b1/b,q,m] b^m),
      {k, 0, Infinity}, m, 2]

CheckRec[%, 1]

(1.10.14) with x->x+1, y->y+1

qZeil[(1-q) qPochhammer[q^(k+1),q,Infinity] q^(k(x+1)) qFactorial[x+y+1,q] /
     (qPochhammer[q^(k+y+1),q,Infinity] qFactorial[x,q] qFactorial[y,q]),
      {k, 0, Infinity}, x, 1, {y}]

Ex. 1.3 (i)

qZeil[qBinomial[n,k,q] b^k qPochhammer[a,q,k] qPochhammer[b,q,n-k] / qPochhammer[a b,q,n],
      {k, 0, n}, n, 1]

Ex. 1.5

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] qPochhammer[c,q,k+m] q^k qPochhammer[b/c,q,n] *
      qPochhammer[b,q,m] /
     (qPochhammer[q,q,k] qPochhammer[c/b q^(1-n),q,k] qPochhammer[c,q,k] qPochhammer[b,q,n] *
      qPochhammer[c,q,m] qPochhammer[b q^n,q,m]), {k, 0, n}, n, 1, {m}]

Ex. 1.6 (i)

qZeil[qPochhammer[q^(-n),q,2k] q^(2k) qPochhammer[b,q,n] q^Binomial[n,2] /
     (qPochhammer[b q,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[b,q^2,n]),
      {k, 0, n}, n, 1]

Ex. 1.7 with a:=q^(-n)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q^(1-n),q,k] z^k /
     (qPochhammer[q^(-n),q,k] qPochhammer[q,q,k] (1+q^(-n) z) *
      qPochhammer[q^(1-2n) z,q,2n-1]), {k, -Infinity, Infinity}, n, 1]

Ex. 1.16 with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q,k] q^(k(k+1)/2) / qPochhammer[q,q,k] /
      qPochhammer[q^(1-2n),q^2,n], {k, 0, 2n}, n, 1]

Ex. 1.17 (LHS with k->n-k)

qZeil[qPochhammer[a,q,n-k] qPochhammer[b,q,n-k] (-a b)^k q^(k(2n-k-1)/2) /
      qPochhammer[q,q,n-k], {k, 0, n}, n, 1]

qZeil[qPochhammer[a,q,n+1] (-b)^k q^Binomial[k,2] /
     (qPochhammer[q,q,k] qPochhammer[q,q,n-k] (1-a q^(n-k))), {k, 0, n}, n, 2]

Ex. 1.20 (i) with x->x+1

qZeil[qPochhammer[q,q,Infinity] (1-q)^(-x) q^((x+1) k) /
     (qPochhammer[q,q,k] qFactorial[x,q]), {k, 0, Infinity}, x, 1]

Ex. 1.20 (ii) with x->x+1

qZeil[(1-q)^x (-1)^k q^((x+1) k) q^Binomial[k,2] qFactorial[x,q] /
     (qPochhammer[q,q,Infinity] qPochhammer[q,q,k]),
      {k, 0, Infinity}, x, 1]

Ex. 1.24

qZeil[qPochhammer[q^(n+1),q,Infinity] (x/2)^n (-x^2 q^(n+1)/4)^k *
      q^(2 Binomial[k,2]) /
     (qPochhammer[q,q,Infinity] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[-x^2/4,q,Infinity] qPochhammer[q^(n+1),q,Infinity] (x/2)^n *
     (-x^2/4)^k / (qPochhammer[q,q,Infinity] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

Ex. 1.25 (i)

qZeil[qPochhammer[q^(n+1),q,Infinity] (x/2)^n (-x^2/4)^k /
     (qPochhammer[q,q,Infinity] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[q^(n+1),q,Infinity] (x/2)^n (-x^2 q^(n+1)/4)^k *
      q^(2 Binomial[k,2]) /
     (qPochhammer[q,q,Infinity] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

Ex. 1.25 (ii), (iii)

qZeil[qPochhammer[q^(n+1),q,Infinity] (x Sqrt[q]/2)^n (-x^2 q/4)^k /
     (qPochhammer[q,q,Infinity] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[q^(n/2) qPochhammer[q^(n+2),q,Infinity] (x/2)^(n+1) (-x^2/4)^k /
     (qPochhammer[q,q,Infinity] qPochhammer[q^(n+2),q,k] qPochhammer[q,q,k]) *
      (x/2 + 2/x (1-q^(n+1+k))), {k, -Infinity, Infinity}, n, 2]

qZeil[q^(-n/2) qPochhammer[q^(n+1),q,Infinity] (x/2)^n (-x^2/4)^k /
     (qPochhammer[q,q,Infinity] qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]) *
      (1 - (1-q^(n+k))), {k, -Infinity, Infinity}, n, 2]

Ex. 1.29 with E^(I θ)->e

qZeil[qPochhammer[b,q,k] qPochhammer[b,q,n-k] e^(n-2k) /
     (qPochhammer[q,q,k] qPochhammer[q,q,n-k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[b,q,n] e^n qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] (q/(b e^2))^k /
     (qPochhammer[q,q,n] qPochhammer[q^(1-n)/b,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[b^2,q,n] (e b)^(-n) qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] *
      qPochhammer[b e^2,q,k] q^k /
     (qPochhammer[q,q,n] qPochhammer[b^2,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

(2.3.2)

qZeil[qPochhammer[a q^(2j),q,k] qPochhammer[q^(j-n),q,k] (-q^(1+n-j))^k *
      qPochhammer[a q^(n+j+1),q,Infinity] qPochhammer[-q^(1+n-j),q,Infinity] /
     (qPochhammer[a q^(j+n+1),q,k] qPochhammer[q,q,k] qPochhammer[-q,q,Infinity] *
      qPochhammer[a q^(2j+1),q^2,Infinity] qPochhammer[a q^(2n+2),q^2,Infinity]),
      {k, -Infinity, Infinity}, n, 1, {j}]

(2.3.4)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[q^(-n),q,k] q^(n k) /
     (qPochhammer[a,q^2,k] qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 0]

(2.7.5) (corrected)

qZeil[qPochhammer[a,q,k] qPochhammer[q^(-n),q,k] (1-a q^(2k)) q^(2k^2+n k) (a^2)^k /
     (qPochhammer[q,q,k] qPochhammer[a q^(n+1),q,k] (1-a)), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a q,q,n] q^(Binomial[k,2]+(n+1)k) (-a)^k /
      qPochhammer[q,q,k], {k, 0, n}, n, 2]

(2.8.1)

qZeil[qPochhammer[λ b c/a,q,k] qPochhammer[λ,q,j] (1 - λ q^(2j)) *
      qPochhammer[λ b/a,q,j] qPochhammer[λ c/a,q,j] qPochhammer[a q/(b c),q,j] *
      qPochhammer[a,q,k+j] qPochhammer[a/λ,q,k-j] (a/λ)^j *
      qPochhammer[q,q,k] qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] /
     (qPochhammer[q a^2/(λ b c),q,k] qPochhammer[q,q,j] (1 - λ) qPochhammer[a q/b,q,j] *
      qPochhammer[a q/c,q,j] qPochhammer[λ b c/a,q,j] qPochhammer[λ q,q,k+j] *
      qPochhammer[q,q,k-j] qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k]),
      {j, 0, k}, k, 1]

(2.10.16) with a:=q^n

qZeil[qPochhammer[q^(1-n),q,k] (q^(1+n)/c)^k / qPochhammer[b q^2/c,q,k],
      {k, 0, n-1}, n, 1]

qZeil[-c (1-b q/c) qPochhammer[q^(-n),q,k] (q^(1+n)/c)^k /
     (q (1-q^n) qPochhammer[b q/c,q,k]), {k, 1, n}, n, 1]

Ex. 2.1 with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q^(2-2n),q^2,k] t^k /
     (qPochhammer[q^(-2n),q^2,k] qPochhammer[q,q,k] (1-q^(1-2n) t^2) *
      qPochhammer[q^(2-2n) t,q,2n-2]), {k, -Infinity, Infinity}, n, 1]

Ex. 2.2 with b= t:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[q^(-n),q,k] q^(-n k) /
     (qPochhammer[a,q^2,k] qPochhammer[a q^(1+n),q,k] qPochhammer[q,q,k]),
     {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[q^n,q,k] qPochhammer[q^(-n),q,k] (a q)^k qPochhammer[q^(-2n),q,n] *
      qPochhammer[a q,q,n] / (qPochhammer[q^(1-2n),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

Ex. 2.6 with n->2n

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[-q^(1-2n)/(b c),q,k] q^k qPochhammer[b,q,2n] qPochhammer[c,q,2n] *
      qPochhammer[b^2 c^2,q^2,n] /
     (qPochhammer[q^(1-2n)/b,q,k] qPochhammer[q^(1-2n)/c,q,k] qPochhammer[-b c,q,k] *
      qPochhammer[q,q,k] qPochhammer[q,q^2,n] qPochhammer[b^2,q^2,n] qPochhammer[c^2,q^2,n] *
      qPochhammer[b c,q,2n]), {k, -Infinity, Infinity}, n, 1]

Ex. 2.8 with b:=q^(-n), n->2n

qZeil[qPochhammer[a,q,k] qPochhammer[q^(-2n),q,k] qPochhammer[c,q^2,k] q^k *
      qPochhammer[a q^(1-2n),q^2,n] qPochhammer[c q,q^2,n] /
     (qPochhammer[a q^(1-2n),q^2,k] qPochhammer[c,q,k] qPochhammer[q,q,k] *
      qPochhammer[q^(1-2n),q^2,n] qPochhammer[c/a q,q^2,n] a^n), {k, 0, 2n}, n, 1]

Ex. 2.10

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a b q^(n+1),q,k] qPochhammer[q^(-x),q,k] *
      qPochhammer[c q^(x-N),q,k] q^k /
     (qPochhammer[a q,q,k] qPochhammer[b c q,q,k] qPochhammer[q^(-N),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2, {x, N}]

qZeil[qPochhammer[a/c q,q,n] qPochhammer[b q,q,n] c^n qPochhammer[q^(-n),q,k] *
      qPochhammer[a b q^(n+1),q,k] qPochhammer[q^(-N+x),q,k] qPochhammer[q^(-x)/c,q,k] *
      q^k /
     (qPochhammer[a q,q,n] qPochhammer[b c q,q,n] qPochhammer[b q,q,k] qPochhammer[a/c q,q,k] *
      qPochhammer[q^(-N),q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2, {x, N}]

Ex. 2.11 (i) with E^(I θ)->e

qZeil[a^(-n) qPochhammer[a b,q,n] qPochhammer[a c,q,n] qPochhammer[a d,q,n] *
      qPochhammer[q^(-n),q,k] qPochhammer[a b c d q^(n-1),q,k] qPochhammer[a e,q,k] *
      qPochhammer[a/e,q,k] q^k /
     (qPochhammer[a b,q,k] qPochhammer[a c,q,k] qPochhammer[a d,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[b^(-n) qPochhammer[a b,q,n] qPochhammer[b c,q,n] qPochhammer[b d,q,n] *
      qPochhammer[q^(-n),q,k] qPochhammer[a b c d q^(n-1),q,k] qPochhammer[b e,q,k] *
      qPochhammer[b/e,q,k] q^k /
     (qPochhammer[a b,q,k] qPochhammer[b c,q,k] qPochhammer[b d,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

Ex. 2.12

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,2k] qPochhammer[a/b,q,k] *
      qPochhammer[a^2/b q^(n+1),q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[a/b q,q,n] qPochhammer[a^2/b q,q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[a^2/b q,q,2k] qPochhammer[b q,q,k] *
      qPochhammer[b/a q^(-n),q,k] qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k] *
      qPochhammer[a q,q,n] qPochhammer[a^2/b^2 q,q,n]),
      {k, -Infinity, Infinity}, n, 2]

CheckRec[%, 1]

Ex. 2.14 (i) with a->a^2

qZeil[qPochhammer[a^2,q,k] qPochhammer[a q,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[1/b^2,q,n] qPochhammer[a^2/b q,q,n] qPochhammer[-a/b,q,n] /
     (qPochhammer[a,q,k] qPochhammer[a^2/b q,q,k] qPochhammer[b^2 q^(1-n),q,k] *
      qPochhammer[q,q,k] qPochhammer[a^2/b^2,q,n] qPochhammer[1/b,q,n] qPochhammer[-a/b q,q,n]),
      {k, 0, n}, n, 1]

Ex. 2.14 (ii) with a->a^2 (corrected)

With[{c = a^4 q / (b d λ)},
qZeil[qPochhammer[a^2,q,k] qPochhammer[a q,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] *
      qPochhammer[q^(-n),q,k] q^k qPochhammer[λ q,q,n] qPochhammer[λ^2/a^4,q,n] *
      qPochhammer[-λ/a,q,n] /
     (qPochhammer[a,q,k] qPochhammer[a^2/b q,q,k] qPochhammer[a^2/c q,q,k] *
      qPochhammer[a^2/d q,q,k] qPochhammer[a^4 q^(1-n)/λ^2,q,k] qPochhammer[q,q,k] *
      qPochhammer[λ/a^2,q,n] qPochhammer[λ^2/a^2,q,n] qPochhammer[-λ q/a,q,n]),
      {k, -Infinity, Infinity}, n, 3]]

With[{c = a^4 q / (b d λ)},
qZeil[qPochhammer[λ,q,k] qPochhammer[λ q^2,q^2,k] qPochhammer[λ b/a^2,q,k] *
      qPochhammer[λ c/a^2,q,k] qPochhammer[λ d/a^2,q,k] qPochhammer[a q,q,k] *
      qPochhammer[-a,q,k] qPochhammer[a^2 q,q^2,k] qPochhammer[λ^2 q^n/a^2,q,k] *
      qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[λ,q^2,k] qPochhammer[a^2 q/b,q,k] qPochhammer[a^2 q/c,q,k] *
      qPochhammer[a^2 q/d,q,k] qPochhammer[λ/a,q,k] qPochhammer[-λ q/a,q,k] *
      qPochhammer[λ^2 q/a^2,q^2,k] qPochhammer[a^2 q^(1-n)/λ,q,k] *
      qPochhammer[λ q^(n+1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]]

Ex. 2.17 (i) with a:=q^(-2n)

With[{b = λ^2 q^(1+2n) / c^2},
qZeil[qPochhammer[λ,q,k] qPochhammer[λ q^2,q^2,k] qPochhammer[q^(-2n),q,k] *
      qPochhammer[b,q,k] qPochhammer[c^2,q^2,k] qPochhammer[λ q/c^2,q,k] (-c^2/λ)^k *
      qPochhammer[b q^(1-2n),q^2,n] qPochhammer[c^2/λ q^(-2n),q,2n] *
      qPochhammer[c^2 q,q^2,n] /
     (qPochhammer[λ,q^2,k] qPochhammer[λ q^(1+2n),q,k] qPochhammer[λ q/b,q,k] *
      qPochhammer[λ^2 q^2/c^2,q^2,k] qPochhammer[c^2,q,k] qPochhammer[q,q,k] *
      qPochhammer[λ q,q,2n] qPochhammer[q^(1-2n),q^2,n] qPochhammer[c^2 q/b,q^2,n]),
      {k, 0, 2n}, n, 1]]

Ex. 2.17 (ii) with a:=q^(-2n)

qZeil[qPochhammer[-c,q,k] qPochhammer[-c q^2,q^2,k] qPochhammer[q^(-2n),q,k] *
      qPochhammer[q^(1+2n),q,k] qPochhammer[c,q,k] qPochhammer[-d,q,k] qPochhammer[-q/d,q,k] *
      c^k qPochhammer[-c q^(-2n),q,2n] qPochhammer[c d q,q^2,n] qPochhammer[c/d q^2,q^2,n] /
     (qPochhammer[-c,q^2,k] qPochhammer[-c q^(1+2n),q,k] qPochhammer[-q^(-2n) c,q,k] *
      qPochhammer[-q,q,k] qPochhammer[c q/d,q,k] qPochhammer[c d,q,k] qPochhammer[q,q,k] *
      qPochhammer[-c q,q,2n] qPochhammer[c d q^(-2n),q^2,n] qPochhammer[c/d q^(1-2n),q^2,n]),
      {k, 0, 2n}, n, 1]

Ex. 2.19

With[{e = a^3 q^(n+2) / (b c d f g)},
qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] qPochhammer[e,q,k] *
      qPochhammer[f,q,k] qPochhammer[g,q,k] qPochhammer[q^(-n),q,k] qPochhammer[a q^2,q^2,k] q^k /
     (qPochhammer[a,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] qPochhammer[a q/d,q,k] *
      qPochhammer[a q/e,q,k] qPochhammer[a q/f,q,k] qPochhammer[a q/g,q,k] *
      qPochhammer[a q^(1+n),q,k] qPochhammer[q,q,k]), {k, -Infinity, Infinity}, n, 2]]

With[{e = a^3 q^(n+2) / (b c d f g)},
qZeil[qPochhammer[e q^(-n)/b,q,k] qPochhammer[e q^(2-n)/b,q^2,k] qPochhammer[e,q,k] *
      qPochhammer[a q/(b c),q,k] qPochhammer[a q/(b d),q,k] qPochhammer[a q/(b f),q,k] *
      qPochhammer[a q/(b g),q,k] qPochhammer[e q^(-n)/a,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[a q,q,n] qPochhammer[a q/(c e),q,n] qPochhammer[a q/(d e),q,n] *
      qPochhammer[a q/(e f),q,n] qPochhammer[a q/(e g),q,n] qPochhammer[b,q,n] e^n /
     (qPochhammer[e q^(-n)/b,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[e q/b,q,k] *
      qPochhammer[q^(1-n)/b,q,k] qPochhammer[c e/a q^(-n),q,k] *
      qPochhammer[d e/a q^(-n),q,k] qPochhammer[e f/a q^(-n),q,k] *
      qPochhammer[e g/a q^(-n),q,k] qPochhammer[q,q,k] qPochhammer[a q/c,q,n] *
      qPochhammer[a q/d,q,n] qPochhammer[a q/e,q,n] qPochhammer[a q/f,q,n] *
      qPochhammer[a q/g,q,n] qPochhammer[b/e,q,n]), {k, -Infinity, Infinity}, n, 2]]

Ex. 2.22 ("i") with e:=q^(-n)

qZeil[qPochhammer[a,q,k] (1-a q^(2k)) qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] *
      qPochhammer[q^(-n),q,k] (-a^2 q^(n+2)/(b c d))^k q^Binomial[k,2] /
     (qPochhammer[q,q,k] (1-a) qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] *
      qPochhammer[a q/d,q,k] qPochhammer[a q^(1+n),q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[a q,q,n] qPochhammer[a q/(b c),q,k] qPochhammer[d,q,k] qPochhammer[q^(-n),q,k] *
      (a q^(1+n)/d)^k /
     (qPochhammer[a q/d,q,n] qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

Ex. 2.22 ("ii") with e:=q^(-n)

qZeil[qPochhammer[a,q,k] (1-a q^(2k)) qPochhammer[d,q,k] qPochhammer[q^(-n),q,k] *
      (-a q^(n+1)/d)^k q^Binomial[k,2] qPochhammer[a q/d,q,n] /
     (qPochhammer[q,q,k] (1-a) qPochhammer[a q/d,q,k] qPochhammer[a q^(1+n),q,k] *
      qPochhammer[a q,q,n]), {k, 0, n}, n, 1]

(3.2.6)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a q^n,q,k] qPochhammer[b,q,k] (d e/(a b))^k /
     (qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a q^n,q,k] qPochhammer[a b q/(d e),q,k] (q/b)^k *
      qPochhammer[a q/d,q,n] qPochhammer[a q/e,q,n] (d e/(a q))^n /
     (qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] qPochhammer[q,q,k] qPochhammer[d,q,n] *
      qPochhammer[e,q,n]), {k, 0, n}, n, 2]

(3.2.11) with d:=q^(-n)

qZeil[qPochhammer[a q/(b c),q,k] qPochhammer[q^(-n),q,k] qPochhammer[e,q,k] *
      (a q^(1+n)/e)^k /
     (qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[a q/e,q,n] qPochhammer[a,q,k] (1-a q^(2k)) qPochhammer[b,q,k] *
      qPochhammer[c,q,k] qPochhammer[q^(-n),q,k] qPochhammer[e,q,k] q^Binomial[k,2] *
      (-a^2 q^(2+n)/(b c e))^k /
     (qPochhammer[a q,q,n] qPochhammer[q,q,k] (1-a) qPochhammer[a q/b,q,k] *
      qPochhammer[a q/c,q,k] qPochhammer[a q^(1+n),q,k] qPochhammer[a q/e,q,k]),
      {k, 0, n}, n, 2]

(3.4.7) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] (q x/b^2)^k /
     (qPochhammer[q^(1-n)/b,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n) x/b,q,k] (1-x/b q^(-n+2k)) qPochhammer[x,q,k] *
      qPochhammer[q^(-n),q,2k] (q x/b^2)^k qPochhammer[q^(1-n) x^2/b^2,q,n] /
     ((1-x/b q^(-n)) qPochhammer[q^(1-n)/b,q,k] qPochhammer[x^2 q^(1-n)/b^2,q,2k] *
      qPochhammer[q,q,k] qPochhammer[q^(1-n) x/b,q,n]),
      {k, -Infinity, Infinity}, n, 2]

(3.4.8) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] (1-q^(-n+2k)) qPochhammer[b,q,k] (x/(q b^2))^k /
     ((1-q^(-n)) qPochhammer[q^(1-n)/b,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n) x/b,q,k] (1-x/b q^(-n+2k)) qPochhammer[x,q,k] *
      qPochhammer[q^(1-n),q,2k] (x/(q b^2))^k qPochhammer[q^(-n) x^2/b^2,q,n-1] /
     ((1-x/b q^(-n)) qPochhammer[q^(1-n)/b,q,k] qPochhammer[x^2 q^(-n)/b^2,q,2k] *
      qPochhammer[q,q,k] qPochhammer[q^(1-n) x/b,q,n-2]),
      {k, -Infinity, Infinity}, n, 2]

(3.5.3) with a:=q^(-n), b^2->b, x^2->x

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[x,q^2,k] (q b)^k /
     (qPochhammer[q^(-2n) x b^2,q^2,k] qPochhammer[q^2,q^2,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[q^(-2n) b x,q^2,n] qPochhammer[q^(-n) b,q,n] qPochhammer[q^(-n) b x/q,q,k] *
      (1-b x q^(2k-n-1)) qPochhammer[q^(-n),q,k] qPochhammer[x,q^2,k] *
      qPochhammer[b x,q^2,k] (b q^(-n))^k /
     (qPochhammer[q^(-2n) b,q^2,n] qPochhammer[q^(-n) b x,q,n] (1-b x q^(-1-n)) *
      qPochhammer[b x q^(-2n),q^2,k] qPochhammer[b^2 x q^(-2n),q^2,k] qPochhammer[b x,q,k] *
      qPochhammer[q,q,k]), {k, -Infinity, Infinity}, n, 2]

(3.5.4) with a:=q^(-n), b^2->b, x^2->x

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[b,q^2,k] (x q^2/b^2)^k /
     (qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n) x/b,q,k] (1-q^(-n+2k) x/b) qPochhammer[q^(-n),q,k] *
      qPochhammer[x,q^2,k] qPochhammer[x q/b,q^2,k] (q^(1-n)/b)^k *
      qPochhammer[q^(1-2n) x/b,q^2,n] qPochhammer[q^(2-2n) x/b^2,q^2,n] /
     ((1-q^(-n) x/b) qPochhammer[q x/b,q,k] qPochhammer[q^(2-2n) x/b^2,q^2,k] *
      qPochhammer[q^(1-2n) x/b,q^2,k] qPochhammer[q,q,k] qPochhammer[q^(1-2n)/b,q,n] *
      qPochhammer[q^(1-n) x/b,q,n]), {k, -Infinity, Infinity}, n, 2]

(3.5.10) with c:=q^(-n)

qZeil[qPochhammer[a,q^2,k] qPochhammer[a q^4,q^4,k] qPochhammer[b,q^2,k] *
      qPochhammer[q^(-n),q,2k] qPochhammer[d,q,2k] (a^2 q^(2+2n)/(b d^2))^k /
     (qPochhammer[a,q^4,k] qPochhammer[a q^2/b,q^2,k] qPochhammer[a q^(1+n),q,2k] *
      qPochhammer[a q/d,q,2k] qPochhammer[q^2,q^2,k]), {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[-a/d,q,k] qPochhammer[-q^2 a/d,q^2,k] qPochhammer[q^(-n),q,k] *
      qPochhammer[b,q^2,k] qPochhammer[a q/d^2,q^2,k] (a q^(1+n)/b)^k *
      qPochhammer[a q,q,n] qPochhammer[a^2 q^2/(d^2 b),q^2,n] /
     (qPochhammer[-a/d,q^2,k] qPochhammer[-a q^(1+n)/d,q,k] *
      qPochhammer[a^2 q^2/(d^2 b),q^2,k] qPochhammer[a q,q^2,k] qPochhammer[q,q,k] *
      qPochhammer[a q/b,q,n] qPochhammer[a^2 q^2/d^2,q^2,n]), {k, 0, n}, n, 2]

(3.6.16)

pqTelescope[(1-a d p^k q^k)(1-b p^k/d q^(-k)) qPochhammer[a,p,k] qPochhammer[b,p,k] *
            qPochhammer[q^(-n),q,k] qPochhammer[a d^2 q^n/b,q,k] q^k /
           ((1-a d)(1-b/d) qPochhammer[d q,q,k] qPochhammer[a d q/b,q,k] *
            qPochhammer[a d p q^n,p,k] qPochhammer[b p/d q^(-n),p,k]), {k, 0, n}]

(3.6.17)

pqTelescope[(1-a p^k q^k)(1-b p^k q^(-k)) qPochhammer[a,p,k] qPochhammer[b,p,k] *
            qPochhammer[q^(-n),q,k] qPochhammer[a q^n/b,q,k] q^k /
           ((1-a)(1-b) qPochhammer[q,q,k] qPochhammer[a q/b,q,k] qPochhammer[a p q^n,p,k] *
            qPochhammer[b p q^(-n),p,k]), {k, 0, n}]

(3.6.18)

pqTelescope[(-1)^(n+j) (1-a p^j q^j)(1-b p^j q^(-j)) qPochhammer[a p q^n,q,n-1] *
            qPochhammer[b p q^(-n),q,n-1] qPochhammer[a p^m q^m,p,j-m] *
            qPochhammer[b p^m q^(-m),p,j-m] (-a q^(1+2m)/b)^(j-m) *
            q^(2 Binomial[j-m,2]) /
           (qPochhammer[q,q,n-j] qPochhammer[a p q^n,p,j] qPochhammer[b p q^(-n),p,j] *
            qPochhammer[b q^(1-2n)/a,q,n-j] qPochhammer[q,q,j-m] *
            qPochhammer[a q^(1+2m)/b,q,j-m]), {j, m, n}]

(3.7.4)

pqTelescope[(1-g p^(r+j) q^(r+j))(1-s p^(r+j) q^(-r-j)) *
            qPochhammer[g p^r q^r,p,j] qPochhammer[s p^r q^(-r),p,j] *
            qPochhammer[q^(-m),q,j] qPochhammer[g/s q^(2r+m),q,j] q^j /
           ((1-g p^r q^r)(1-s p^r q^(-r)) qPochhammer[q,q,j] *
            qPochhammer[g/s q^(2r+1),q,j] qPochhammer[g p^(r+1) q^(r+m),p,j] *
            qPochhammer[s p^(r+1) q^(-r-m),p,j]), {j, 0, m}, {r}]

(3.8.1) with p:=q (corrected)

qZeil[(1-a d q^(3k)) (1-b/(d q^k)) qPochhammer[a,q,k] qPochhammer[b,q,k] *
      qPochhammer[q^(-2n),q^2,k] qPochhammer[a d^2/b q^(2n),q^2,k] q^(2k) /
     ((1-a d) (1-b/d) qPochhammer[d q^2,q^2,k] qPochhammer[a d/b q^2,q^2,k] *
      qPochhammer[a d q^(2n+1),q,k] qPochhammer[b/d q^(1-2n),q,k]),
      {k, 0, n}, n, 0]

(3.8.13) with a:=q^(-2n), k->n-k

With[{a = q^(-2n)},
qZeil[qPochhammer[a,q^2,k] (1-a q^(3k)) qPochhammer[d,q^2,k] qPochhammer[a q/d,q^2,k] *
      qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[a q/(b c),q,k] q^k /
     (qPochhammer[q,q,k] qPochhammer[a q/d,q,k] qPochhammer[d,q,k] qPochhammer[a q^2/b,q^2,k] *
      qPochhammer[a q^2/c,q^2,k] qPochhammer[b c q,q^2,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 1]]

(3.8.16)

qZeil[(1-a c q^(4k)) (1-b/c q^(-2k)) qPochhammer[a,q,k] qPochhammer[b,q,k] *
      qPochhammer[q^(-3n),q^3,k] qPochhammer[a c^2/b q^(3n),q^3,k] q^(3k) /
     ((1-a c) (1-b/c) qPochhammer[c q^3,q^3,k] qPochhammer[a c/b q^3,q^3,k] *
      qPochhammer[a c q^(3n+1),q,k] qPochhammer[b/c q^(1-3n),q,k]),
      {k, 0, n}, n, 0]

(3.8.20)

qZeil[(1-a c q^(5k)) (q^(3k)-b/c) qPochhammer[a,q,k] qPochhammer[b,q,k] *
      qPochhammer[q^(-4n),q^4,k] qPochhammer[a c^2/b q^(4n),q^4,k] q^k /
     ((1-a c) (1-b/c) qPochhammer[c q^4,q^4,k] qPochhammer[a c/b q^4,q^4,k] *
      qPochhammer[a c q^(4n+1),q,k] qPochhammer[b/c q^(1-4n),q,k]),
      {k, 0, n}, n, 0]

(3.10.3) with b^2->b, c^2->c

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[-a q^2,q^2,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[-a/w q,q,k] qPochhammer[q^(-n),q,k] (a w q^(n+2)/(b c))^k /
     (qPochhammer[-a,q^2,k] qPochhammer[a^2/b q^2,q^2,k] qPochhammer[a^2/c q^2,q^2,k] *
      qPochhammer[q^2,q^2,k] qPochhammer[w,q,k] qPochhammer[-a q^(n+1),q,k]),
      {k, 0, n}, n, 3]

qZeil[qPochhammer[a,q,2k] qPochhammer[a^2/w^2 q^2,q^2,k] qPochhammer[a^2/(b c) q^2,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] q^(2k) qPochhammer[w/a,q,n] qPochhammer[-a q,q,n] /
     (qPochhammer[a^2/b q^2,q^2,k] qPochhammer[a^2/c q^2,q^2,k] *
      qPochhammer[a/w q^(1-n),q,2k] qPochhammer[q^2,q^2,k] qPochhammer[w,q,n] qPochhammer[-q,q,n]),
      {k, -Infinity, Infinity}, n, 3]

(3.10.4) with x^2->x, y^2->y

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[x,q^2,k] *
      qPochhammer[y,q^2,k] qPochhammer[q^(-2n),q^2,k] (-a^3 q^(2n+3)/(b x y))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a/b q,q,k] qPochhammer[a^2/x q^2,q^2,k] *
      qPochhammer[a^2/y q^2,q^2,k] qPochhammer[a^2 q^(2n+2),q^2,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[x,q^2,k] qPochhammer[y,q^2,k] *
      qPochhammer[-a/b q,q,2k] q^(2k) qPochhammer[a^2 q^2,q^2,n] *
      qPochhammer[a^2/(x y) q^2,q^2,n] /
     (qPochhammer[x y/a^2 q^(-2n),q^2,k] qPochhammer[a^2/b^2 q^2,q^2,k] *
      qPochhammer[-a q,q,2k] qPochhammer[q^2,q^2,k] qPochhammer[a^2/x q^2,q^2,n] *
      qPochhammer[a^2/y q^2,q^2,n]), {k, -Infinity, Infinity}, n, 3]

(3.10.5) (corrected, simplified)

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[-a/w q,q,k] *
      qPochhammer[q^(-n),q,k] (w q^(n-1)/a)^k qPochhammer[-q,q,n] qPochhammer[w,q,n] *
      qPochhammer[a/w q,q,n] /
     (qPochhammer[a^2,q^4,k] qPochhammer[q^2,q^2,k] qPochhammer[w,q,k] qPochhammer[-a q^(n+1),q,k] *
      qPochhammer[-a q,q,n] qPochhammer[a/w q^2,q,n] qPochhammer[w/(a q),q,n]),
      {k, 0, n}, n, 1]

(3.10.6)

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[-a q^2,q^2,k] qPochhammer[b^2,q^2,k] *
      qPochhammer[-a/b^2 q^n,q,k] qPochhammer[q^(-n),q,k] q^(2k) *
      qPochhammer[a/b^2 q^n,q,n] qPochhammer[-q,q,n] qPochhammer[q/b^2,q^2,n] *
      qPochhammer[a^2/b^2 q^2,q^2,n] /
     (qPochhammer[-a,q^2,k] qPochhammer[a^2/b^2 q^2,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[b^2 q^(1-n),q,k] qPochhammer[-a q^(n+1),q,k] qPochhammer[q^n/b^2,q,n] *
      qPochhammer[-a q,q,n] qPochhammer[a/b^2 q,q^2,n] qPochhammer[a/b^2 q^2,q^2,n]),
      {k, 0, n}, n, 1]

(3.10.7) (corrected)

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[b^2,q^2,k] *
      qPochhammer[-a/b^2 q^n,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[-q,q,n] qPochhammer[1/b^2,q,n] qPochhammer[a^2/b^2 q^2,q^2,n] /
     (qPochhammer[a^2,q^4,k] qPochhammer[a^2/b^2 q^2,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[b^2 q^(1-n),q,k] qPochhammer[-a q^(n+1),q,k] qPochhammer[-a q,q,n] *
      qPochhammer[a/b^2,q,n] qPochhammer[1/b^2,q^2,n] q^n), {k, 0, n}, n, 1]

(3.10.8) with b^2->b

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[b,q^2,k] *
      qPochhammer[-a/b q^(n-1),q,k] qPochhammer[q^(-n),q,k] q^(2k) qPochhammer[-q,q,n] *
      qPochhammer[1/(b q),q,n] qPochhammer[a/(b q),q^2,n] qPochhammer[a^2/b q^2,q^2,n] /
     (qPochhammer[a^2,q^4,k] qPochhammer[a^2/b q^2,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[b q^(2-n),q,k] qPochhammer[-a q^(n+1),q,k] qPochhammer[-a q,q,n] *
      qPochhammer[a/(b q),q,n] qPochhammer[a/b q,q^2,n] qPochhammer[1/(b q^2),q^2,n] q^n),
      {k, -Infinity, Infinity}, n, 1]

(3.10.9) with a->a^2 (simplified)

qZeil[qPochhammer[a^2,q,k] qPochhammer[a q,q,k] qPochhammer[-w/(a q),q,k] qPochhammer[q^(-n),q,k] *
      q^k qPochhammer[-1/a,q,n] qPochhammer[a^2/w q,q,n] qPochhammer[w,q,n] /
     (qPochhammer[a,q,k] qPochhammer[w,q,k] qPochhammer[q,q,k] qPochhammer[-a q^(1-n),q,k] *
      qPochhammer[w/(q a^2),q,n] qPochhammer[-a,q,n] qPochhammer[q^2 a^2/w,q,n]),
      {k, 0, n}, n, 1]

(3.10.10) with a->a^2

qZeil[qPochhammer[a^2,q,k] qPochhammer[b,q,k] qPochhammer[-b q^(1-n),q,k] qPochhammer[q^(-n),q,k] *
      q^k (1+q^n/b) (1+a/b) qPochhammer[a^2/b q,q,n] qPochhammer[a/b,q,n] *
      qPochhammer[1/b^2,q,n] /
     (qPochhammer[a^2/b q,q,k] qPochhammer[b^2 q^(1-n),q,k] qPochhammer[-b q^(-n),q,k] *
      qPochhammer[q,q,k] (1+1/b) (1+a/b q^n) qPochhammer[a^2/b^2,q,n] *
      qPochhammer[a/b q,q,n] qPochhammer[1/b,q,n]), {k, 0, n}, n, 1]

(3.10.11)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[-w/b,q,k] qPochhammer[q^(-n),q,k] q^k /
      (qPochhammer[a/b q,q,k] qPochhammer[-b q^(-n),q,k] qPochhammer[w,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[a,q^2,k] qPochhammer[a^2/w^2 q^2,q^2,k] qPochhammer[a/b^2 q,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] q^(2k) qPochhammer[w/a,q,n] qPochhammer[-a q/b,q,n] /
     (qPochhammer[a^2/b^2 q^2,q^2,k] qPochhammer[a/w q^(1-n),q,2k] *
      qPochhammer[q^2,q^2,k] qPochhammer[w,q,n] qPochhammer[-q/b,q,n]), {k, 0, n}, n, 2]

(3.10.12)

qZeil[qPochhammer[a,q,k] qPochhammer[a/b^2 q,q,k] qPochhammer[-a q/w,q,k] qPochhammer[q^(-n),q,k] q^k /
      (qPochhammer[a/b q,q,k] qPochhammer[-a/b q,q,k] qPochhammer[a/w q^(1-n),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[a,q^2,k] qPochhammer[a^2/w^2 q^2,q^2,k] qPochhammer[a/b^2 q,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (qPochhammer[a^2/b^2 q^2,q^2,k] qPochhammer[a/w q^(1-n),q,2k] qPochhammer[q^2,q^2,k]),
      {k, 0, n}, n, 2]

(3.10.14) (RHS simplified)

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[b^2,q^2,k] *
      qPochhammer[c^2,q^2,k] qPochhammer[-a/w q,q,k] qPochhammer[q^(-n),q,k] *
      (a w q^(n+1)/(b^2 c^2))^k /
     (qPochhammer[a^2,q^4,k] qPochhammer[a^2/b^2 q^2,q^2,k] qPochhammer[a^2/c^2 q^2,q^2,k] *
      qPochhammer[q^2,q^2,k] qPochhammer[w,q,k] qPochhammer[-a q^(n+1),q,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a q,q,2k] qPochhammer[a^2/(b^2 c^2) q^2,q^2,k] *
      qPochhammer[a^2/w^2 q^2,q^2,k] qPochhammer[q^(-2n),q^2,k] q^(2k) *
      qPochhammer[-a q,q,n] qPochhammer[a/w q^2,q,n] qPochhammer[w/(a q),q,n] /
     (qPochhammer[a^2/b^2 q^2,q^2,k] qPochhammer[a^2/c^2 q^2,q^2,k] *
      qPochhammer[a/w q^(2-n),q,2k] qPochhammer[q^2,q^2,k] qPochhammer[-q,q,n] *
      qPochhammer[a/w q,q,n] qPochhammer[w,q,n]), {k, -Infinity, Infinity}, n, 3]

(3.10.15) with b^2->b, c^2->c, d^2->d (corrected)

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[a^2,q^2,k] qPochhammer[-a q^2,q^2,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[d,q^2,k] qPochhammer[-λ q^(n+1)/a,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[-a,q^2,k] qPochhammer[a^2/b q^2,q^2,k] qPochhammer[a^2/c q^2,q^2,k] *
      qPochhammer[a^2/d q^2,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[a^2 q^(-n)/λ,q,k] *
      qPochhammer[-a q^(n+1),q,k]), {k, -Infinity, Infinity}, n, 3]]

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[-a q,q,n] qPochhammer[λ q/a,q,n] qPochhammer[λ q^2/a^2,q^2,n] *
      qPochhammer[λ,q^2,k] qPochhammer[λ q^4,q^4,k] qPochhammer[a,q,2k] *
      qPochhammer[λ b/a^2,q^2,k] qPochhammer[λ c/a^2,q^2,k] qPochhammer[λ d/a^2,q^2,k] *
      qPochhammer[λ^2 q^(2n+2)/a^2,q^2,k] qPochhammer[q^(-2n),q^2,k] *
      (q a^2/λ)^k /
     (qPochhammer[-q,q,n] qPochhammer[λ q/a^2,q,n] qPochhammer[λ q^2,q^2,n] *
      qPochhammer[λ,q^4,k] qPochhammer[λ q/a,q,2k] qPochhammer[a^2 q^2/b,q^2,k] *
      qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[a^2 q^2/d,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[a^2 q^(-2n)/λ,q^2,k] qPochhammer[λ q^(2n+2),q^2,k]),
      {k, -Infinity, Infinity}, n, 3]]

(3.10.16) with b^2->b, c^2->c, d^2->d (corrected)

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[d,q^2,k] qPochhammer[-λ q^n/a,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a^2,q^4,k] qPochhammer[a^2/b q^2,q^2,k] qPochhammer[a^2/c q^2,q^2,k] *
      qPochhammer[a^2/d q^2,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[a^2 q^(1-n)/λ,q,k] *
      qPochhammer[-a q^(n+1),q,k]), {k, -Infinity, Infinity}, n, 3]]

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[-a q,q,n] qPochhammer[λ/a,q,n] qPochhammer[λ/a^2,q^2,n] q^n *
      qPochhammer[λ,q^2,k] qPochhammer[λ q^4,q^4,k] qPochhammer[a q,q,2k] *
      qPochhammer[λ b/a^2,q^2,k] qPochhammer[λ c/a^2,q^2,k] qPochhammer[λ d/a^2,q^2,k] *
      qPochhammer[λ^2 q^(2n)/a^2,q^2,k] qPochhammer[q^(-2n),q^2,k] *
      (q a^2/λ)^k /
     (qPochhammer[-q,q,n] qPochhammer[λ/a^2,q,n] qPochhammer[λ q^2,q^2,n] *
      qPochhammer[λ,q^4,k] qPochhammer[λ/a,q,2k] qPochhammer[a^2 q^2/b,q^2,k] *
      qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[a^2 q^2/d,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[a^2 q^(2-2n)/λ,q^2,k] qPochhammer[λ q^(2n+2),q^2,k]),
      {k, -Infinity, Infinity}, n, 3]]

Ex. 3.1 with z:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a b q,q^2,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[a,q^2,k] qPochhammer[b,q^2,k] qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (qPochhammer[a b q,q^2,k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 2]

Ex. 3.2 (i) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b^2,q^2,k] (-z)^k /
     (qPochhammer[b^2,q,k] qPochhammer[z q^(-n),q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,2k] (z^2)^k /
     (qPochhammer[q b^2,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[z q^(-n),q,n]),
      {k, -Infinity, Infinity}, n, 2]

Ex. 3.2 (ii) with b:=q^(-n)

qZeil[qPochhammer[a^2,q,k] qPochhammer[a^2 q^(-2n),q^2,k] z^k /
     (qPochhammer[a^2 q^(-2n),q,k] qPochhammer[-z a^2,q,k] qPochhammer[q,q,k]),
      {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[q^(-2n),q^2,k] (-a^2 z^2 q)^k *
      q^(2 Binomial[k,2]) /
     (qPochhammer[a^2 q^(1-2n),q^2,k] qPochhammer[a^2 z^2,q^2,k] qPochhammer[q^2,q^2,k]),
      {k, 0, Infinity}, n, 2]

Ex. 3.3 with z:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[q/a,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[c,q,k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[-q^(1-n)/c,q,n] qPochhammer[c/a,q^2,k] qPochhammer[a c/q,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (qPochhammer[-q^(-n),q,n] qPochhammer[c^2,q^2,k] qPochhammer[q^2,q^2,k]),
      {k, 0, n}, n, 2]

Ex. 3.4 ("i")

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b^2,q^2,k] q^(n k-Binomial[k,2]) /
     (qPochhammer[q,q,k] qPochhammer[b^2,q,k]), {k, 0, n}, n, 2]

Ex. 3.4 ("ii")

qZeil[qPochhammer[q^(-n),q,2k] qPochhammer[a,q,2k] q^(2k) /
     (qPochhammer[b^2 q,q^2,k] qPochhammer[d,q,2k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 3]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,k] qPochhammer[b^2,q^2,k] (1/d)^k *
      q^(-Binomial[k,2]+k) a^n qPochhammer[d/a,q,n] /
     (qPochhammer[b^2,q,k] qPochhammer[a/d q^(1-n),q,k] qPochhammer[q,q,k] qPochhammer[d,q,n]),
      {k, 0, n}, n, 3]

Ex. 3.9 with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[l q,q,k] qPochhammer[b,q,k] (l^2 q^n / b^2)^k *
      (1-l) (1+l/b) qPochhammer[q l^2/b,q,n]/
     (qPochhammer[l,q,k] qPochhammer[q l^2/b,q,k] qPochhammer[q,q,k] (1-l+l/b(1-l q^n)) *
      qPochhammer[l^2/b^2,q,n]), {k, 0, n}, n, 1]

Ex. 3.10 with a:=q^(-n)

qZeil[qPochhammer[-l,q,k] qPochhammer[l^2 q^4,q^4,k] qPochhammer[q^(-n),q,k] *
      qPochhammer[b^2,q^2,k] (l q^n/b^2)^k qPochhammer[l q/b,q,n] *
      qPochhammer[-l q/b,q,n] qPochhammer[l,q,n] /
     (qPochhammer[l^2,q^4,k] qPochhammer[l^2 q^2/b^2,q^2,k] qPochhammer[-l q^(n+1),q,k] *
      qPochhammer[q,q,k] qPochhammer[l q,q,n] qPochhammer[-l q,q,n] qPochhammer[l/b^2,q,n]),
      {k, 0, n}, n, 1]

Ex. 3.12 with b^2->b, c^2->c, d->d q^(-n) (LHS out of memory, RHS with k->n-k)

qZeil[qPochhammer[q^(-2n),q^2,k] (1+q^(2k-n)) qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[d,q,k] qPochhammer[-q^(1-n)/w,q,k] (w q^(2-n)/(b c d))^k /
     ((1+q^(-n)) qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^(2-2n)/c,q^2,k] *
      qPochhammer[q^2,q^2,k] qPochhammer[-q^(1-n)/d,q,k] qPochhammer[w,q,k]) /. d->d q^(-n),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[-1,q,n] qPochhammer[w/d,q,n] d^n qPochhammer[d^2,q^2,k] *
      qPochhammer[q^(2-2n)/(b c),q^2,k] qPochhammer[q^(2-2n)/w^2,q^2,k] *
      qPochhammer[q^(-n),q,2k] q^(2k) /
     (qPochhammer[-d,q,n] qPochhammer[w,q,n] qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[q^(2-2n)/c,q^2,k] qPochhammer[d q^(1-n)/w,q,2k]) /. {k->n-k, d->d q^(-n)},
      {k, -Infinity, Infinity}, n, 3]

Ex. 3.13 ("i") with b^2->b, c^2->c, d^2->d (corrected)

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[a^2,q^2,k] qPochhammer[-a q^2,q^2,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[d,q^2,k] qPochhammer[-λ q^n/a,q,k] qPochhammer[q^(-n),q,k] q^(2k) /
     (qPochhammer[-a,q^2,k] qPochhammer[a^2/b q^2,q^2,k] qPochhammer[a^2/c q^2,q^2,k] *
      qPochhammer[a^2/d q^2,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[a^2 q^(1-n)/λ,q,k] *
      qPochhammer[-a q^(n+1),q,k]), {k, -Infinity, Infinity}, n, 3]]

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[-a q,q,n] qPochhammer[λ/a,q,n] qPochhammer[λ/a^2,q^2,n] *
      qPochhammer[λ q^2/a,q^2,n] qPochhammer[λ,q^2,k] qPochhammer[λ q^4,q^4,k] *
      qPochhammer[a,q,2k] qPochhammer[λ b/a^2,q^2,k] qPochhammer[λ c/a^2,q^2,k] *
      qPochhammer[λ d/a^2,q^2,k] qPochhammer[λ^2 q^(2n)/a^2,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] (a^2 q^3/λ)^k /
     (qPochhammer[-q,q,n] qPochhammer[λ/a^2,q,n] qPochhammer[λ q^2,q^2,n] *
      qPochhammer[λ/a,q^2,n] qPochhammer[λ,q^4,k] qPochhammer[λ q/a,q,2k] *
      qPochhammer[a^2 q^2/b,q^2,k] qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[a^2 q^2/d,q^2,k] *
      qPochhammer[q^2,q^2,k] qPochhammer[a^2 q^(2-2n)/λ,q^2,k] *
      qPochhammer[λ q^(2n+2),q^2,k]), {k, -Infinity, Infinity}, n, 3]]

Ex. 3.13 ("ii") with b^2->b, c^2->c, d^2->d (corrected)

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[d,q^2,k] qPochhammer[-λ q^(n-1)/a,q,k] qPochhammer[q^(-n),q,k] q^(2k) /
     (qPochhammer[a^2,q^4,k] qPochhammer[a^2/b q^2,q^2,k] qPochhammer[a^2/c q^2,q^2,k] *
      qPochhammer[a^2/d q^2,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[a^2 q^(2-n)/λ,q,k] *
      qPochhammer[-a q^(n+1),q,k]), {k, -Infinity, Infinity}, n, 3]]

With[{c = a^4 q^2 / (b d λ)},
qZeil[qPochhammer[-a q,q,n] qPochhammer[λ/(a q),q,n] qPochhammer[λ/(a^2 q^2),q^2,n] *
      (1-λ q^(2n-1)/a) q^n qPochhammer[λ,q^2,k] qPochhammer[λ q^4,q^4,k] *
      qPochhammer[a q,q,2k] qPochhammer[λ b/a^2,q^2,k] qPochhammer[λ c/a^2,q^2,k] *
      qPochhammer[λ d/a^2,q^2,k] qPochhammer[λ^2 q^(2n-2)/a^2,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] (a^2 q^3/λ)^k /
     (qPochhammer[-q,q,n] qPochhammer[λ/(a^2 q),q,n] qPochhammer[λ q^2,q^2,n] (1-λ/(a q)) *
      qPochhammer[λ,q^4,k] qPochhammer[λ/a,q,2k] qPochhammer[a^2 q^2/b,q^2,k] *
      qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[a^2 q^2/d,q^2,k] qPochhammer[q^2,q^2,k] *
      qPochhammer[a^2 q^(4-2n)/λ,q^2,k] qPochhammer[λ q^(2n+2),q^2,k]),
      {k, -Infinity, Infinity}, n, 3]]

Ex. 3.14 (corrected)

qZeil[(-x^2 q^(n+1)/4)^k q^(2 Binomial[k,2]) /
     (qPochhammer[q^(n+1),q,k] qPochhammer[q,q,k]), {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[-x^2 q^n/4,q,k] (1+x^2 q^(2k+n)/4) qPochhammer[-x^2/4,q,k] *
      (-x^2 q^(2n)/4)^k q^(2k^2) /
     (qPochhammer[-x^2 q^(n+1)/4,q,Infinity] qPochhammer[q,q,k] (1+x^2 q^n/4) *
      qPochhammer[q^(n+1),q,k]), {k, 0, Infinity}, n, 2]

Ex. 3.15 with q->q^2, I->i

qZeil[(-x^2 q^(2n+2)/4)^k q^(4 Binomial[k,2]) /
     (qPochhammer[q^(2n+2),q^2,k] qPochhammer[q^2,q^2,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[i x/2,q,Infinity] qPochhammer[q^(2n+1),q^2,k] (i x/2)^k /
     (qPochhammer[q^(2n+1),q,k] qPochhammer[q,q,k]),
      {k, 0, Infinity}, n, 2] /. i -> I

Ex. 3.20 ("ii") with p:=q, a:=q^n

qZeil[(1-q^(2k+n)) qPochhammer[q^n,q,k] qPochhammer[c/b,q,k] b^k /
     ((1-q^n) qPochhammer[q,q,k] qPochhammer[b q^(1+n),q,k]), {k, 0, Infinity}, n, 2]

qZeil[(1-c) qPochhammer[q^(1+n),q,k] qPochhammer[c/b,q,k] (b q)^k /
     ((1-b) qPochhammer[q,q,k] qPochhammer[b q^(1+n),q,k]), {k, 0, Infinity}, n, 2]

qZeil[(1-c) qPochhammer[q^(1+n),q,k] qPochhammer[c/b q,q,k] b^k /
     ((1-b q^(1+n)) qPochhammer[q,q,k] qPochhammer[b q^(2+n),q,k]),
      {k, 0, Infinity}, n, 2]

qZeil[(1-c) qPochhammer[q^(n+1),q,Infinity] qPochhammer[b,q,k] *
      qPochhammer[c q^(k+1),q,Infinity] q^((n+1)k) /
     ((1-b) qPochhammer[b q^(n+1),q,Infinity] qPochhammer[q,q,k] *
      qPochhammer[b q^(k+1),q,Infinity]), {k, 0, Infinity}, n, 2]

Ex. 3.31 (i)

qZeil[(1-a c q^(4k)) qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[c/b q,q,2k] *
      qPochhammer[a^2 b c q^(3n),q^3,k] qPochhammer[q^(-3n),q^3,k] q^k /
     ((1-a c) qPochhammer[c q^3,q^3,k] qPochhammer[a c/b q^3,q^3,k] *
      qPochhammer[a b,q,2k] qPochhammer[q^(1-3n)/(a b),q,k] qPochhammer[a c q^(3n+1),q,k]),
      {k, 0, n}, n, 1]

qZeil[(1-a b/q^2) (1-a b q^(3n)) (1-a c q^(3n)) (1-a c q^(6k+2)) *
      qPochhammer[a q^2,q^3,k] qPochhammer[b q^2,q^3,k] qPochhammer[c/b q^2,q^3,k] *
      qPochhammer[c/b q^3,q^3,k] qPochhammer[a^2 b c q^(3n),q^3,k] *
      qPochhammer[q^(-3n),q^3,k] q^(3k) /
     ((1-a c q^(3n+2)) (1-a b q^(3n-2)) (1-a b) (1-a c) *
      qPochhammer[c q^3,q^3,k] qPochhammer[a c/b q^3,q^3,k] qPochhammer[a b q^3,q^3,k] *
      qPochhammer[a b q^2,q^3,k] qPochhammer[q^(5-3n)/(a b),q^3,k] *
      qPochhammer[a c q^(3n+5),q^3,k]), {k, 0, n}, n, 1]

Ex. 5.5 with z:=q^(2n)

qZeil[q^(4k^2+4n k) (1+q^(4k+1+2n)) /
     (qPochhammer[-q^(2n+1),q^2,Infinity] qPochhammer[-q^(1-2n),q^2,Infinity]),
      {k, -Infinity, Infinity}, n, 1]

Ex. 5.11 ("i") (RHS 180 sec)

qZeil[qPochhammer[q/c,q,n] qPochhammer[q/d,q,n] qPochhammer[a q/e,q,n] qPochhammer[a q/f,q,n] *
      qPochhammer[a q^2,q^2,k] qPochhammer[c,q,k] qPochhammer[d,q,k] qPochhammer[e,q,k] qPochhammer[f,q,k] *
      qPochhammer[a q^(-n),q,k] qPochhammer[q^(-n),q,k] (a^2 q^(2n+2)/(c d e f))^k /
     (qPochhammer[a q,q,n] qPochhammer[q/a,q,n] qPochhammer[a q/(c d),q,n] *
      qPochhammer[a q/(e f),q,n] qPochhammer[a,q^2,k] qPochhammer[a q/c,q,k] *
      qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] qPochhammer[a q/f,q,k] qPochhammer[q^(n+1),q,k] *
      qPochhammer[a q^(n+1),q,k]), {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[e,q,k] qPochhammer[f,q,k] qPochhammer[a q^(n+1)/(c d),q,k] *
      qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a q/c,q,k] qPochhammer[a q/d,q,k] qPochhammer[q^(n+1),q,k] *
      qPochhammer[e f/(a q^n),q,k]), {k, -Infinity, Infinity}, n, 2]

Ex. 5.11 ("ii") with f:=q^(-n), c:=a q^(-n)

qZeil[qPochhammer[e,q,k] qPochhammer[q^(-n),q,k] (a q^(n+1)/e)^k /
     (qPochhammer[q^(n+1),q,k] qPochhammer[a q/d,q,k]), {k, -n, n}, n, 2]

qZeil[(1-a q^(2k)) qPochhammer[a q^(-n),q,k] qPochhammer[d,q,k] qPochhammer[e,q,k] *
      qPochhammer[q^(-n),q,k] (a^2 q^(1+2n)/(d e))^k q^(k^2) *
      qPochhammer[q/d,q,n] qPochhammer[a q/e,q,n] /
     ((1-a) qPochhammer[q^(n+1),q,k] qPochhammer[a q/d,q,k] qPochhammer[a q/e,q,k] *
      qPochhammer[a q^(n+1),q,k] qPochhammer[q/a,q,n] qPochhammer[a q,q,n]),
      {k, -Infinity, Infinity}, n, 2]

Ex. 5.11 ("iii") with a:=q^n

qZeil[q^(k^2+n k) / qPochhammer[q,q,k], {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[q^n,q,k] (1-q^(n+2k)) (-1)^k q^(k(5k-1)/2+2n k) /
     (qPochhammer[q^(n+1),q,Infinity] qPochhammer[q,q,k] (1-q^n)),
      {k, 0, Infinity}, n, 2]

Ex. 5.18 (ii) with b:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] (q^(2+n)/(c d))^k *
      qPochhammer[q^2/c,q,n] qPochhammer[q^2/d,q,n] /
     (qPochhammer[q^(2+n),q,k] qPochhammer[q^2/c,q,k] qPochhammer[q^2/d,q,k] qPochhammer[q,q,n+1] *
      qPochhammer[q^2/(c d),q,n]), {k, -n-1, n}, n, 1, MagicFactor->-k-1]

Ex. 5.18 (iii) (corrected)

With[{e = q^(n+1)/(b c d)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] *
      qPochhammer[e,q,k] q^k qPochhammer[q/b,q,n] qPochhammer[q/c,q,n] qPochhammer[q/d,q,n] *
      qPochhammer[q/(b c d),q,n] /
     (qPochhammer[q^(1+n),q,k] qPochhammer[q/b,q,k] qPochhammer[q/c,q,k] qPochhammer[q/d,q,k] *
      qPochhammer[q/e,q,k] qPochhammer[q,q,n] qPochhammer[q/(b c),q,n] *
      qPochhammer[q/(b d),q,n] qPochhammer[q/(c d),q,n]),
      {k, -n, n}, n, 1, MagicFactor->-k]]

Ex. 5.18 (iv) (corrected)

With[{e = q^(n+3)/(b c d)},
qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] *
      qPochhammer[e,q,k] q^k qPochhammer[q^2/b,q,n] qPochhammer[q^2/c,q,n] *
      qPochhammer[q^2/d,q,n] qPochhammer[q^2/(b c d),q,n] /
     (qPochhammer[q^(2+n),q,k] qPochhammer[q^2/b,q,k] qPochhammer[q^2/c,q,k] *
      qPochhammer[q^2/d,q,k] qPochhammer[q^2/e,q,k] (1-q) qPochhammer[q^2,q,n] *
      qPochhammer[q^2/(b c),q,n] qPochhammer[q^2/(b d),q,n] qPochhammer[q^2/(c d),q,n]),
      {k, -n-1, n}, n, 1, MagicFactor->-k-1]]

Ex. 5.19 (i)

qZeil[(-1)^k qBinomial[2n,n+k,q]^3 q^(k(3k+1)/2) qPochhammer[q,q,n]^3 /
      qPochhammer[q,q,3n], {k, -n, n}, n, 1, MagicFactor->-k]

Ex. 5.19 (ii)

qZeil[(-1)^k qBinomial[2n+1,n+k+1,q]^3 q^(k(3k+1)/2) qPochhammer[q,q,n]^3 /
      qPochhammer[q,q,3n+1], {k, -n-1, n}, n, 1, MagicFactor->-k-1]

Ex. 5.20 (i) with a:=q^(-n), c:=q^(n+1)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q,k] z^k / (qPochhammer[q^(1+n),q,k] qPochhammer[d,q,k]),
      {k, -n, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[q^(-n) b z/d,q,k] (d q^n)^k *
      qPochhammer[q^(-n) z,q,n] qPochhammer[d q^(1+n)/(b z),q,n] /
     (qPochhammer[q^(-n) z,q,k] qPochhammer[q^(1+n),q,k] qPochhammer[d,q,n] qPochhammer[q/b,q,n]),
      {k, -Infinity, Infinity}, n, 2]

(7.2.4)

qZeil[qPochhammer[a b q,q,k] qPochhammer[a b q^3,q^2,k] qPochhammer[a q,q,k] qPochhammer[c q,q,k] *
      qPochhammer[q^(-n),q,k] (b/c q^n)^k qPochhammer[b q,q,n] qPochhammer[a b/c q,q,n] /
     (qPochhammer[a b q,q^2,k] qPochhammer[b q,q,k] qPochhammer[a b/c q,q,k] *
      qPochhammer[a b q^(n+2),q,k] qPochhammer[q,q,k] qPochhammer[a b q^2,q,n] *
      qPochhammer[b/c,q,n]), {k, 0, n}, n, 1]

(7.2.11) / (7.2.1)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a b q^(n+1),q,k] qPochhammer[q^(-j),q,k] *
      qPochhammer[c d q^(j+1),q,k] q^k /
     (qPochhammer[a q,q,k] qPochhammer[c q,q,k] qPochhammer[b d q,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

(7.3.6)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a b q^(n+1),q,k] (x q)^k /
     (qPochhammer[a q,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

(7.4.14) with E^(I θ)->e

qZeil[qPochhammer[b,q,k] qPochhammer[b,q,n-k] e^(n-2k) /
      (qPochhammer[q,q,k] qPochhammer[q,q,n-k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[b^2,q,n] b^(-n/2) qPochhammer[q^(-n),q,k] qPochhammer[b^2 q^n,q,k] *
      qPochhammer[Sqrt[b] e,q,k] qPochhammer[Sqrt[b]/e,q,k] q^k /
     (qPochhammer[q,q,n] qPochhammer[b^2 q,q^2,k] qPochhammer[-b,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

(7.5.3) with E^(I θ)->e

qZeil[qPochhammer[a b,q,n] qPochhammer[a c,q,n] qPochhammer[a d,q,n] a^(-n) *
      qPochhammer[q^(-n),q,k] qPochhammer[a b c d q^(n-1),q,k] qPochhammer[a e,q,k] *
      qPochhammer[a/e,q,k] q^k /
     (qPochhammer[a b,q,k] qPochhammer[a c,q,k] qPochhammer[a d,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

(7.5.26) with E^(I θ)->e, q^α->a, q^β->b, q->q^2

qZeil[qPochhammer[a^2 q^4,q^4,n] qPochhammer[q^(-4n),q^4,k] *
      qPochhammer[a^2 b^2 q^(4n+4),q^4,k] qPochhammer[a q e,q^4,k] *
      qPochhammer[a q/e,q^4,k] q^(4k) /
     (qPochhammer[q^4,q^4,n] qPochhammer[a^2 q^4,q^4,k] qPochhammer[-a b q^2,q^2,2k] *
      qPochhammer[q^4,q^4,k]), {k, -Infinity, Infinity}, n, 2]

qZeil[a^n qPochhammer[a q^2,q^2,n] qPochhammer[-b q^2,q^2,n] qPochhammer[q^(-2n),q^2,k] *
      qPochhammer[a b q^(2n+2),q^2,k] qPochhammer[q e,q^2,k] qPochhammer[q/e,q^2,k] q^(2k) /
     (qPochhammer[-a b q^2,q^2,n] qPochhammer[q^2,q^2,n] qPochhammer[a q^2,q^2,k] *
      qPochhammer[-b q^2,q^2,k] qPochhammer[q^4,q^4,k]),
      {k, -Infinity, Infinity}, n, 2]

(7.5.33)  with E^(I θ)->e, q^λ->l, q->q^2

qZeil[qPochhammer[l,q^2,k] qPochhammer[l,q^2,n-k] e^(n-2k) /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,n-k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[l^2,q^2,n] l^(-n/2) qPochhammer[l q,q^2,n] qPochhammer[q^(-2n),q^2,k] *
      qPochhammer[l^2 q^(2n),q^2,k] qPochhammer[l^(1/2) e,q^2,k] *
      qPochhammer[l^(1/2)/e,q^2,k] q^(2k) /
     (qPochhammer[l q,q^2,n] qPochhammer[q^2,q^2,n] qPochhammer[l q,q^2,k] *
      qPochhammer[-l,q^2,k] qPochhammer[-l q,q^2,k] qPochhammer[q^2,q^2,k]),
      {k, -Infinity, Infinity}, n, 2]

(7.5.40)  with E^(I θ)->e, q^λ->l, q->q^2

qZeil[qPochhammer[l,q^2,k] qPochhammer[l,q^2,2n-k] e^(2n-2k) /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,2n-k]), {k, 0, 2n}, n, 2]

qZeil[qPochhammer[l,q^2,n] qPochhammer[l q,q^2,n] q^(-n) qPochhammer[q^(-2n),q^2,k] *
      qPochhammer[l q^(2n),q^2,k] qPochhammer[q e^2,q^2,k] qPochhammer[q/e^2,q^2,k] q^(2k) /
     (qPochhammer[q^2,q^2,n] qPochhammer[q,q^2,n] qPochhammer[l q,q^2,k] *
      qPochhammer[-q,q^2,k] qPochhammer[q^4,q^4,k]), {k, 0, n}, n, 2]

Ex. 7.8 (i)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[x,q,k] qPochhammer[-q^n/a,q,k] q^k /
     (qPochhammer[q^(-N),q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2, {N}]

(8.1.2)

qZeil[qPochhammer[q,q,j] qPochhammer[q,q,k] qPochhammer[a,q,j+k] qPochhammer[q^(-x),q,m] *
      qPochhammer[a q^x,q,m] q^(Binomial[j,2]+Binomial[k,2]+Binomial[m+1,2]-
      m(j+k)) (-1)^(j+k+m) /
     (qPochhammer[a,q,m] qPochhammer[q,q,m-j] qPochhammer[q,q,m-k] qPochhammer[q,q,j+k-m] *
      qPochhammer[q^(-x),q,j] qPochhammer[a q^x,q,j] qPochhammer[q^(-x),q,k] qPochhammer[a q^x,q,k]),
      {m, -Infinity, j+k}, j, 1, {x}]

(8.4.5) with E^(I ϕ)->e1, E^(I θ)->e2, q->q^2

qZeil[qPochhammer[a^2 e1^2,q^4,k] qPochhammer[a e2,q^2,k] qPochhammer[a/e2,q^2,k] *
      qPochhammer[a e1 q,q^2,k] qPochhammer[-a^3 e1/q,q^2,k] qPochhammer[-a^3 e1 q^3,q^4,k] *
      qPochhammer[a^4 q^(2n),q^2,k] qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (qPochhammer[-a^3 e1/q,q^4,k] qPochhammer[-a^2,q^2,k] qPochhammer[a^4 q^2,q^4,k] *
      qPochhammer[-a^2 e1 q/e2,q^2,k] qPochhammer[-a^2 e1 e2 q,q^2,k] *
      qPochhammer[-e1 q^(1-2n)/a,q^2,k] qPochhammer[-a^3 e1 q^(1+2n),q^2,k] *
      qPochhammer[q^2,q^2,k]), {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[a^2 e1^2,q^2,k] qPochhammer[-q e1 e2,q^2,k] qPochhammer[-q e1/e2,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] q^(2k) qPochhammer[a^2/e1^2,q^2,n] *
      qPochhammer[-a^3 q e1,q^2,n] /
     (qPochhammer[-a^2 q e1/e2,q^2,k] qPochhammer[-a^2 q e1 e2,q^2,k] *
      qPochhammer[q^(2-2n) e1^2/a^2,q^2,k] qPochhammer[q^2,q^2,k] qPochhammer[a^4,q^2,n] *
      qPochhammer[-a q/e1,q^2,n]), {k, 0, n}, n, 2]

(8.7.9) with b:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[q^(-n),q,k] (w/(a q))^k /
      (qPochhammer[a,q^2,k] qPochhammer[w,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[q^(1-n),q,k] (w q^n)^k *
      qPochhammer[w/a q^(-1-n),q,n] /
     (qPochhammer[w/a q^(-n),q,k] qPochhammer[q,q,k] qPochhammer[w,q,n]),
      {k, 0, n}, n, 2]

Ex. 8.16 with b:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[q,q^2,k] qPochhammer[-q,q,k] *
      qPochhammer[a^2 q^n,q,k] qPochhammer[a^2/c,q,k] qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] *
      (q/a)^k /
     (qPochhammer[a,q^2,k] qPochhammer[a^2 q,q^2,k] qPochhammer[-a,q,k] qPochhammer[q^(1-n)/a,q,k] *
      qPochhammer[c q/a,q,k] qPochhammer[a q^(1+n),q,k] qPochhammer[a q/c,q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a q,q,n] qPochhammer[a^2/c,q,n] qPochhammer[a,q,k] qPochhammer[q^(-n),q,k] *
      qPochhammer[c,q,k] qPochhammer[q^(-n) c/a,q,k] (q^2/a^2)^k /
     (qPochhammer[a^2,q,n] qPochhammer[a q/c,q,n] qPochhammer[q^(1-n)/a,q,k] qPochhammer[c q/a,q,k] *
      qPochhammer[q^(1-n) c/a^2,q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

Gessel, Stanton [1983]

(1.4) with q->q^2 (corrected)

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[a d^2 q^(2n+1),q^2,k] qPochhammer[a/d^2,q^2,k] *
      qPochhammer[a,q,k] qPochhammer[q^2 a/c,q,k] qPochhammer[c/(a q),q,k] (1-a q^(3k)) q^k *
      qPochhammer[d^2 q,q,2n] qPochhammer[c,q^2,n] qPochhammer[a^2 q^3/c,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[a q^(2n+1),q,k] qPochhammer[q^(-2n)/d^2,q,k] *
      qPochhammer[d^2 q,q,k] qPochhammer[c,q^2,k] qPochhammer[a^2 q^3/c,q^2,k] (1-a) *
      qPochhammer[a q,q,2n] qPochhammer[c d^2/a,q^2,n] qPochhammer[d^2 a q^3/c,q^2,n]),
      {k, 0, n}, n, 1]

(2.18) (corrected)

qZeil[qPochhammer[q^(1-a-(b+1)n),q,n-k-1] qPochhammer[q^(a+(b+1)l),q,k-l] *
      q^(-(k-l)(n+n b+a)) / (qPochhammer[q,q,n-k] qPochhammer[q,q,k-l]) *
      ((1-q^(-a-b l-l))(1-q^(n-l))-(1-q^((n-l)(b+1)))(1-q^(k-l))),
      {k, l, n}, n, 0, {a, b}]

Theorem 3.1 with q->q^2 (modified)

qZeil[qPochhammer[a q^(3l),q,k-l] q^(-2k l) qPochhammer[a q,q,3n-1] *
      (1-a q^(3k)) q^(2(Binomial[n-k+1,2]+k n)) (-1)^(n-k) /
     (qPochhammer[q^2,q^2,k-l] qPochhammer[q^2,q^2,n-k] qPochhammer[a q,q,2n+k]),
      {k, l, n}, n, 0]

Theorem 3.2

qZeil[qPochhammer[a q^(2k),q,n-k] q^(-k n) qPochhammer[a q,q,2k-1] *
      (1-a q^(2l)) q^(Binomial[k-l+1,2]+k l) (-1)^(k-l) /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,k-l] qPochhammer[a q,q,k+l]), {k, l, n}, n, 0]

Theorem 3.3

qZeil[qPochhammer[a q^k,q,n-k] q^(-k n) qPochhammer[a q^l,q,k-l] *
      q^(Binomial[k-l+1,2]+l k) (-1)^(k-l) /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,k-l]), {k, l, n}, n, 0]

Theorem 3.4 with q->q^2 (modified)

qZeil[qPochhammer[a q^l,1/q,k-l] q^(-2k l) qPochhammer[a q^(n+1),q,n-k-1] *
      (1-a q^k) q^(2(Binomial[n-k+1,2]+k n)) (-1)^(n-k) /
     (qPochhammer[q^2,q^2,k-l] qPochhammer[q^2,q^2,n-k]), {k, l, n}, n, 0]

Theorem 3.5 with q->q^3 (modified)

qZeil[qPochhammer[a q^(2l),1/q,k-l] q^(-3k l) qPochhammer[a q^(2n+1),q,n-k-1] *
      (1-a q^(2k)) q^(3(Binomial[n-k+1,2]+k n)) (-1)^(n-k) /
     (qPochhammer[q^3,q^3,k-l] qPochhammer[q^3,q^3,n-k]), {k, l, n}, n, 0]

Theorem 3.6 with q->q^2 (modified)

qZeil[qPochhammer[a q^(3l),q^2,k-l] q^(-k l) qPochhammer[a q^(3n-2),1/q^2,n-k-1] *
      (1-a q^(3k)) q^(Binomial[n-k+1,2]+k n) (-1)^(n-k) /
     (qPochhammer[q,q,k-l] qPochhammer[q,q,n-k]), {k, l, n}, n, 0]

Theorem 3.7 (modified)

pqTelescope[qPochhammer[a p^l q^l,p,k-l] qPochhammer[a q^n p^(n-1),1/p,n-k-1] *
            (1-a p^k q^k) q^(Binomial[n-k+1,2]+k n) (-1)^(n-k) q^(-k l) /
            (qPochhammer[q,q,k-l] qPochhammer[q,q,n-k]), {k, l, n}]

(4.6) with b:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[a q/c,q,k] qPochhammer[c q^n,q,k] (q^(-n) x)^k /
     (qPochhammer[c,q,k] qPochhammer[a q^(1-n)/c,q,k] qPochhammer[q,q,k]),
      {k, 0, Infinity}, n, 3]

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[a,q,2k] q^k /
     (qPochhammer[q,q,k] qPochhammer[c,q,k] qPochhammer[a q^(1-n)/c,q,k] qPochhammer[a x,q,k] *
      qPochhammer[q/x,q,k]), {k, 0, n}, n, 3]

(4.7) (corrected)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[a q/c,q,k] qPochhammer[c/b,q,k] *
      qPochhammer[q^(-n),q,k] (b q^n)^k qPochhammer[c,q,n] qPochhammer[q a b/c,q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[c,q,k] qPochhammer[q a b/c,q,k] qPochhammer[a q^(n+1),q,k] *
      qPochhammer[q,q,k] qPochhammer[b,q,n] qPochhammer[a q,q,n]), {k, 0, n}, n, 1]

(4.12) with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[q^(1-2n)/c,q^2,k] (c x^2)^k /
     (qPochhammer[c q,q^2,k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[c,q^2,k] (-x)^k q^(-Binomial[k,2]) *
      qPochhammer[q^(-2n) x,q,2n] qPochhammer[x,q,-k] /
     (qPochhammer[q,q,k] qPochhammer[c,q,k] qPochhammer[q^(-2n) x,q,k]),
      {k, -Infinity, Infinity}, n, 2]

(4.13) with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q^2,k] (-q^(1-2n) x^2)^k q^(2 Binomial[k,2]) /
      qPochhammer[q^2,q^2,k], {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-2n),q^2,k] (-x)^k q^(-Binomial[k,2]) qPochhammer[x,q,-k] *
      qPochhammer[q^(-2n) x,q,2n] /
     (qPochhammer[q,q,k] qPochhammer[q^(-2n) x,q,k]), {k, 0, n}, n, 2]

(4.14) with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q^2,k] q^(-k^2) (-x^2)^k / qPochhammer[q^2,q^2,k],
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-2n),q^2,k] (-x)^k qPochhammer[x,q,-k] qPochhammer[q^(-2n) x,q,2n] /
     (qPochhammer[q,q,k] qPochhammer[q^(-2n) x,q,k]), {k, 0, n}, n, 2]

(4.21) with q->q^2, a->a^2, b:=q^(-n)

qZeil[qPochhammer[a,q,k] qPochhammer[q^(-n),q,k] (1+a) (-q^(1+n) a x)^k /
     (qPochhammer[q,q,k] qPochhammer[a q^(1+n),q,k] (1+a q^(2k))),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a^2,q^2,2k] qPochhammer[-a q^(1+n),q,2k] q^(-2Binomial[k,2]) *
      (-x)^k qPochhammer[x,q^2,-k] /
     (qPochhammer[q^2,q^2,k] qPochhammer[a^2 q^(2+2n),q^2,k] qPochhammer[a^2 x,q^2,k] *
      qPochhammer[-q a,q,2k]), {k, -Infinity, Infinity}, n, 3]

(4.22) with q->q^2, a->a^2 (inversion trick)

qZeil[(-1)^k q^(2(Binomial[k,2]+n)) qPochhammer[a^2 q^2,q^2,k-1] *
      (1-a^2 q^(4k)) qPochhammer[-q,q,k] qPochhammer[b,q,k] (1+a) (-a q/b)^k *
      qPochhammer[a^2 q^2/b^2,q^2,n] qPochhammer[-a q,q,2n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,n-k] qPochhammer[a^2 q^2,q^2,n+k] *
      qPochhammer[-a,q,k] qPochhammer[a q/b,q,k] (1+a q^(2k)) *
      qPochhammer[-a q/b,q,2n] q^(2n)), {k, 0, n}, n, 1]

(4.25) with q->q^3, a->a^3, x:=-q^(3n)

qZeil[qPochhammer[a,q,k] (1-a q^(2k)) (-a q^(3n+1))^k /
     (qPochhammer[q,q,k] (1-a^3 q^(6k))), {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[-a^3 q^(3n),q^3,Infinity] qPochhammer[a,q,3k+1] qPochhammer[-q^(3n),q^3,-k] *
      q^(-3Binomial[k,2]+3n k) /
     (qPochhammer[-q^(3n),q^3,Infinity] qPochhammer[q^3,q^3,k] (1-a^3 q^(6k)) *
      qPochhammer[-a^3 q^(3n),q^3,k]), {k, -Infinity, Infinity}, n, 3]

(4.26) with q->q^3, a->a^3 (inversion trick)

qZeil[(-1)^k q^(3(Binomial[k,2]+n)) qPochhammer[a q,q,k-1] (1-a q^(2k)) *
      (a q)^k qPochhammer[q^3,q^3,n] qPochhammer[a^3 q^3,q^3,2n] /
     (qPochhammer[q^3,q^3,n-k] qPochhammer[a^3 q^3,q^3,n+k] qPochhammer[q,q,k] *
      qPochhammer[a q,q,3n] q^(3n)), {k, 0, n}, n, 1]

(4.30) with a:=q^(-3n)

qZeil[qPochhammer[q^(-3n),q^3,k] q^(-Binomial[k,2]) (-x)^k qPochhammer[x,q,-k] *
      qPochhammer[q^(-3n) x,q,3n] /
     (qPochhammer[q,q,k] qPochhammer[q^(-3n) x,q,k] qPochhammer[q^(-6n) x^3,q^3,n]),
      {k, 0, n}, n, 1]

(4.32) with n->3n

qZeil[qPochhammer[q^(-3n),q,k] qPochhammer[a q^(3n),q,k] qPochhammer[a,q^3,k] q^k *
      qPochhammer[a,q,3n] qPochhammer[q^3,q^3,n] /
     (qPochhammer[q,q,k] qPochhammer[a,q,2k] qPochhammer[a,q^3,n] qPochhammer[q,q,3n] a^n),
      {k, 0, 3n}, n, 1]

(5.6) with a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[q^(1-2n),q^2,k] (c x^2)^k /
     (qPochhammer[c q,q^2,k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[c,q^2,k] (-x)^k q^(-Binomial[k,2]) *
      qPochhammer[q^(-2n) x,q,2n] qPochhammer[x,q,-k] /
     (qPochhammer[q,q,k] qPochhammer[c,q,k]), {k, 0, 2n}, n, 2]

(5.9) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[c,q^2,k] x^k q^(-Binomial[k,2]) /
     (qPochhammer[c,q,k] qPochhammer[q,q,k]), {k, 0, n}, n, 3]

qZeil[qPochhammer[q^(-n),q,2k] (x^2)^k q^(-Binomial[2k,2]) qPochhammer[x,q,-2k] *
      qPochhammer[q^(-n) x,q,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[c q,q^2,k]),
      {k, -Infinity, Infinity}, n, 3]

(5.13) with q->q^2, a:=q^(-2n) (corrected)

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[c,q,2k] (-x)^k q^(-k^2) /
     (qPochhammer[c^2,q^2,k] qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 3]

qZeil[qPochhammer[q^(-2n),q^2,k] (-x)^k q^(k-2 Binomial[k+1,2]) *
      qPochhammer[x,q^2,-k] qPochhammer[q^(-2n) x,q^2,n] /
     (qPochhammer[q,q,k] qPochhammer[-c,q,k]), {k, 0, n}, n, 3]

(5.14) with q->q^2 (inversion trick)

qZeil[(-1)^k q^(2(Binomial[k,2]-k(n-1))) qPochhammer[-q,q,k] q^(-k) *
      qPochhammer[q^2,q^2,n] qPochhammer[c^2,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,n-k] qPochhammer[-c,q,k] qPochhammer[c,q,2n] *
      q^(-n^2) (-1)^n), {k, 0, n}, n, 1]

(5.15) with q->q^2 (inversion trick)

qZeil[(-1)^k q^(2(Binomial[k,2]+n)) qPochhammer[-q,q,k] c^k qPochhammer[c^2,q^2,n] *
      qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,n-k] qPochhammer[-c,q,k] qPochhammer[c,q,2n] *
      q^(2n)), {k, 0, n}, n, 1]

(5.18) with q->q^2, a:=q^(-2n)

qZeil[qPochhammer[q^(-2n),q^2,k] (c x)^k / (qPochhammer[-c,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[c,q,2k] (-x)^k q^(-2 Binomial[k,2]) *
      qPochhammer[x,q^2,-k] qPochhammer[q^(-2n) x,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[c^2,q^2,k]), {k, 0, n}, n, 2]

(5.19) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b,q^2,k] y^k / (qPochhammer[b,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 3]

qZeil[qPochhammer[q^(-n),q,2k] (b y^2)^k q^(4 Binomial[k,2]+k) *
      qPochhammer[q^(-n) y,q,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[b q,q^2,k] qPochhammer[q^(-n) y,q,2k]),
      {k, -Infinity, Infinity}, n, 3]

(6.3) with q->q^2

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[b/a,q,k] qPochhammer[a q/b,q,k] q^k *
      qPochhammer[a,q,2n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^(1-2n)/a,q,k] qPochhammer[a,q,k] qPochhammer[b,q^2,n] *
      qPochhammer[a^2 q/b,q^2,n]), {k, 0, n}, n, 1]

(6.8) with q->q^2

qZeil[qPochhammer[q^(-2n),q^2,k] qPochhammer[a,q,k] qPochhammer[a q^2/c,q,k] *
      qPochhammer[c/(a q),q,k] (1-a q^(3k)) q^(2n k+Binomial[k+1,2]) a^k *
      qPochhammer[c,q^2,n] qPochhammer[a^2 q^3/c,q^2,n] /
     (qPochhammer[a q^(2n+1),q,k] qPochhammer[q^2,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[a^2 q^3/c,q^2,k] (1-a) qPochhammer[a q,q,2n]),
      {k, 0, n}, n, 1]

(6.13) with q->q^2, n->2n (corrected)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[a,q^2,k] qPochhammer[b/q,q^2,k] *
      qPochhammer[q^2 a/b,q^2,k] (1-a q^(3k)) q^(2n k-Binomial[k,2]) *
      qPochhammer[b,q^2,n] qPochhammer[a q^3/b,q^2,n] /
     (qPochhammer[q,q,k] qPochhammer[a q^(2+2n),q^2,k] qPochhammer[b/q,q,k] *
      qPochhammer[q^2 a/b,q,k] (1-a) qPochhammer[a q^2,q^2,n] qPochhammer[q,q^2,n]),
      {k, 0, 2n}, n, 1]

(6.14) with q->q^2, n->2n (corrected)

qZeil[qPochhammer[q^(-2n),q,k] qPochhammer[q^2 a/f,q,k] qPochhammer[f q^(2n-1),q,k] *
      qPochhammer[a,q^2,k] qPochhammer[q^2 a/b,q^2,k] qPochhammer[b/q,q^2,k] (1-a q^(3k)) *
      q^k qPochhammer[f,q^2,n] qPochhammer[a q^3/b,q^2,n] qPochhammer[b,q^2,n] *
      qPochhammer[f/(a q),q^2,n] /
     (qPochhammer[a q^(2+2n),q^2,k] qPochhammer[f,q^2,k] qPochhammer[a q^(3-2n)/f,q^2,k] *
      qPochhammer[q,q,k] qPochhammer[b/q,q,k] qPochhammer[q^2 a/b,q,k] qPochhammer[a q^2,q^2,n] *
      (1-a) qPochhammer[f q/b,q^2,n] qPochhammer[b f/(a q^2),q^2,n] qPochhammer[q,q^2,n]),
      {k, 0, 2n}, n, 1]

(6.20) with q->q^3 (corrected)

qZeil[qPochhammer[q^(-3n),q^3,k] qPochhammer[b/q,q,k] qPochhammer[q^2/b,q,k] q^k *
      qPochhammer[q,q,3n] /
     (qPochhammer[q,q,2k] qPochhammer[q^(-3n),q,k] qPochhammer[b,q^3,n] *
      qPochhammer[q^3/b,q^3,n] qPochhammer[q^3,q^3,n]), {k, 0, n}, n, 1]

(6.24) with q->q^3

qZeil[qPochhammer[q^(-3n),q^3,k] qPochhammer[q/b,q,k] qPochhammer[b q,q,k] q^k *
      qPochhammer[q^2,q,3n] /
     (qPochhammer[q^2,q,2k] qPochhammer[q^(-3n-1),q,k] qPochhammer[b q^3,q^3,n] *
      qPochhammer[q^3,q^3,n] qPochhammer[q^3/b,q^3,n]), {k, 0, n}, n, 1]

(6.28) with q->q^3 (inversion trick)

qZeil[qPochhammer[1/a,q,k] (1-a q^(2k)) qPochhammer[a,q,3n-k] (-a q)^k *
      q^(3n+k^2-3Binomial[k+1,2]) qPochhammer[q^3,q^3,n] qPochhammer[a^3,q^3,n] /
     (qPochhammer[q^3,q^3,n-k] qPochhammer[1/q,1/q,k] qPochhammer[1/a,1/q,k] *
      qPochhammer[a q^(1-k),q,k] (1-a q^(2k)) qPochhammer[a^3,q^3,2n] q^(3n)),
      {k, 0, n}, n, 1]

(7.3) with x:=q^n

qZeil[q^(2k^2-k+2n k) / qPochhammer[q^2,q^2,k], {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[q^n,q,Infinity] q^(Binomial[k,2]+n k) /
     (qPochhammer[q,q,k] qPochhammer[q^n,q,k]), {k, 0, Infinity}, n, 2]

(7.6) with x:=q^(2n)

qZeil[q^(k^2+2n k) / qPochhammer[q,q,k], {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[-q^(2n+2),q^2,Infinity] q^(k^2+2n k) /
     (qPochhammer[q^2,q^2,k] qPochhammer[-q^(2n+2),q^2,k]),
      {k, 0, Infinity}, n, 2]

(7.9) with y:=q^n

qZeil[qPochhammer[b,q^2,k] q^(Binomial[k,2]+n k) (-1)^k /
     (qPochhammer[b,q,k] qPochhammer[q,q,k]), {k, 0, Infinity}, n, 3]

qZeil[qPochhammer[q^n,q,Infinity] q^(4k^2-2k+2n k) b^k /
     (qPochhammer[q^2,q^2,k] qPochhammer[b q,q^2,k] qPochhammer[q^n,q,2k]),
      {k, 0, Infinity}, n, 3]

(7.17) with q->q^2, x:=q^(2n)

qZeil[q^(2k^2+k+2n k) / qPochhammer[q^2,q^2,k], {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[q^(2+2n),q^2,Infinity] q^((3k^2+k)/2+2n k) /
     (qPochhammer[q,q,k] qPochhammer[q^(2+2n),q^2,k]), {k, 0, Infinity}, n, 2]

(7.20) with x:=q^n

qZeil[(-1)^k q^(Binomial[k,2]+n k) / (qPochhammer[q,q,k] qPochhammer[c,q,k]),
      {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[q^n,q,Infinity] q^((3k^2-3k)/2+n k) (-c)^k /
     (qPochhammer[q,q,k] qPochhammer[c,q,k] qPochhammer[q^n,q,k]),
      {k, 0, Infinity}, n, 2]

(7.23) with x:=q^(2n) (corrected)

qZeil[q^(k^2+2n k) / (qPochhammer[q,q,k] qPochhammer[q/b,q,k]),
      {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[-q^(2+2n),q^2,Infinity] qPochhammer[-q/b,q,2k] q^(k^2+2n k) /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2/b^2,q^2,k] qPochhammer[-q^(2+2n),q^2,k]),
      {k, 0, Infinity}, n, 2]

Jain [1981]

(3.1) with a:=q^(-n)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b^2,q^2,k] (-z)^k /
     (qPochhammer[b^2,q,k] qPochhammer[q^(-n) z,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[q^(-n),q,2k] z^(2k) /
     (qPochhammer[q^(-n) z,q,n] qPochhammer[b^2 q,q^2,k] qPochhammer[q^2,q^2,k]),
      {k, 0, n}, n, 2]

(3.8) with z:=q^n

qZeil[qPochhammer[a^2,q,k] qPochhammer[a^2 b^2,q^2,k] q^(n k) /
     (qPochhammer[a^2 b^2,q,k] qPochhammer[-q^n a^2,q,k] qPochhammer[q,q,k]),
      {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[a^2 q^(2n),q^2,Infinity] qPochhammer[a^2,q^2,k] qPochhammer[b^2,q^2,k] *
      (a^2 q^(1+2n))^k (-1)^k q^(2Binomial[k,2]) /
     (qPochhammer[-q^n a^2,q,Infinity] qPochhammer[q^n,q,Infinity] *
      qPochhammer[a^2 b^2 q,q^2,k] qPochhammer[q^(2n) a^2,q^2,k] qPochhammer[q^2,q^2,k]),
      {k, 0, Infinity}, n, 2]

(3.18)

qZeil[qPochhammer[a,q,k] qPochhammer[q/a,q,k] qPochhammer[c,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[-q,q,k] qPochhammer[b,q,k] qPochhammer[-c/b q^(1-n),q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2]

qZeil[qPochhammer[-b,q,n] qPochhammer[-q/c,q,n] qPochhammer[b/a,q^2,k] qPochhammer[a b/q,q^2,k] *
      qPochhammer[c^2,q^2,k] qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (qPochhammer[-b/c,q,n] qPochhammer[-q,q,n] qPochhammer[b^2,q^2,k] qPochhammer[-c q^(-n),q,2k] *
      qPochhammer[q^2,q^2,k]), {k, 0, n}, n, 2]

(3.19)

qZeil[qPochhammer[a,q,k] qPochhammer[q/a,q,k] qPochhammer[q^(-2n),q^2,k] q^k qPochhammer[b,q,2n] /
     (qPochhammer[-q,q,k] qPochhammer[b,q,k] qPochhammer[q^(1-2n)/b,q,k] qPochhammer[q,q,k] *
      qPochhammer[a b,q^2,n] qPochhammer[b q/a,q^2,n]), {k, 0, n}, n, 1]

Jain, Verma [1980]

(1.1) (RHS with k->n-k)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[q^(-4n),q^4,k] *
      (a q^(1+4n))^k q^(-Binomial[k,2]) /
     (qPochhammer[a,q^2,k] qPochhammer[a^4 q^(4+4n),q^4,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[a q,q,4n] qPochhammer[q^(-4n),q^4,k] q^(2k) /
     (qPochhammer[a^4 q^(4+4n),q^4,n] qPochhammer[q^(-4n)/a,q,2k] qPochhammer[q^2,q^2,k]) /.
      k->n-k, {k, -Infinity, Infinity}, n, 2]

(1.2) with e^2->e (RHS with k->n-k)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[c,q,k] qPochhammer[e,q^2,k] *
      qPochhammer[q^(-2n),q^2,k] (a^2 q^(2+2n)/(c e))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a q/c,q,k] qPochhammer[a^2 q^2/e,q^2,k] *
      qPochhammer[a^2 q^(2+2n),q^2,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[a^2 q^2,q^2,n] qPochhammer[-a q/e,q,2n] *
      qPochhammer[a^2 q^2/(c^2 e),q^2,n] e^n qPochhammer[c e q^(-1-2n)/a^2,q,2k] *
      qPochhammer[e,q^2,k] qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (qPochhammer[-a q,q,2n] qPochhammer[a^2 q^2/c^2,q^2,n] qPochhammer[a^2 q^2/e,q^2,n] *
      qPochhammer[e c^2 q^(-2n)/a^2,q^2,k] qPochhammer[-e q^(-2n)/a,q,2k] *
      qPochhammer[q^2,q^2,k]) /. k->n-k, {k, -Infinity, Infinity}, n, 2]

(1.4)

qZeil[qPochhammer[a,q^2,k] qPochhammer[a q^4,q^4,k] qPochhammer[b,q^2,k] qPochhammer[x,q,2k] *
      qPochhammer[y,q,2k] qPochhammer[q^(-n),q,2k] (a^3 q^(3+2n)/(b x^2 y^2))^k /
     (qPochhammer[a,q^4,k] qPochhammer[a q^2/b,q^2,k] qPochhammer[a q/x,q,2k] *
      qPochhammer[a q/y,q,2k] qPochhammer[a q^(1+n),q,2k] qPochhammer[q^2,q^2,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a q,q,n] qPochhammer[a q/(x y),q,n] qPochhammer[x,q,k] qPochhammer[y,q,k] *
      qPochhammer[a q/b,q^2,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a q/x,q,n] qPochhammer[a q/y,q,n] qPochhammer[a q,q^2,k] qPochhammer[a q/b,q,k] *
      qPochhammer[x y q^(-n)/a,q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

(1.5)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[x,q^3,k] qPochhammer[y,q^3,k] *
      qPochhammer[q^(-3n),q^3,k] (a^4 q^(4+3n)/(x y))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a^3 q^3/x,q^3,k] qPochhammer[a^3 q^3/y,q^3,k] *
      qPochhammer[a^3 q^(3+3n),q^3,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 4]

qZeil[qPochhammer[a^3 q^3,q^3,n] qPochhammer[a^3 q^3/(x y),q^3,n] qPochhammer[a q,q,3k] *
      qPochhammer[x,q^3,k] qPochhammer[y,q^3,k] qPochhammer[q^(-3n),q^3,k] q^(3k) /
     (qPochhammer[a^3 q^3/x,q^3,n] qPochhammer[a^3 q^3/y,q^3,n] *
      qPochhammer[a^3 q^3,q^3,2k] qPochhammer[x y q^(-3n)/a^3,q^3,k] qPochhammer[q^3,q^3,k]),
      {k, -Infinity, Infinity}, n, 4]

(1.6)

qZeil[qPochhammer[a,q^3,k] qPochhammer[a q^6,q^6,k] qPochhammer[x,q,3k] qPochhammer[y,q,3k] *
      qPochhammer[q^(-n),q,3k] (a^4 q^(3+3n)/(x^3 y^3))^k /
     (qPochhammer[a,q^6,k] qPochhammer[a q/x,q,3k] qPochhammer[a q/y,q,3k] *
      qPochhammer[a q^(1+n),q,3k] qPochhammer[q^3,q^3,k]),
      {k, -Infinity, Infinity}, n, 4]

qZeil[qPochhammer[a q,q,n] qPochhammer[a q/(x y),q,n] qPochhammer[a,q^3,k] qPochhammer[x,q,k] *
      qPochhammer[y,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a q/x,q,n] qPochhammer[a q/y,q,n] qPochhammer[a,q,2k] *
      qPochhammer[x y q^(-n)/a,q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 4]

(1.7) (corrected)

qZeil[qPochhammer[a,q^3,k] (1-a q^(6k)) qPochhammer[x,q,3k] qPochhammer[y,q,3k] *
      qPochhammer[q^(-n),q,3k] (a^4 q^(3n)/(x^3 y^3))^k (1-a/x q^(3k) -
      a/y q^(3k) + a q^(6k) - a q^(n+3k) + a^2/(x y) q^(n+3k)) /
     (qPochhammer[q^3,q^3,k] (1-a) qPochhammer[a q/x,q,3k] *
      qPochhammer[a q/y,q,3k] qPochhammer[a q^(1+n),q,3k]),
      {k, -Infinity, Infinity}, n, 4]

qZeil[qPochhammer[a q,q,n] qPochhammer[a/(x y),q,n] qPochhammer[a,q^3,k] qPochhammer[x,q,k] *
      qPochhammer[y,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a q/x,q,n-1] qPochhammer[a q/y,q,n-1] qPochhammer[a,q,2k] *
      qPochhammer[x y/a q^(1-n),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 4]

(1.8) (corrected)

qZeil[qPochhammer[a,q^2,k] qPochhammer[a q^4,q^4,k] qPochhammer[b,q^2,k] qPochhammer[x,q,2k] *
      qPochhammer[y,q,2k] qPochhammer[q^(-n),q,2k] (a^3 q^(1+2n)/(b x^2 y^2))^k *
      (1-a/x q^(2k) - a/y q^(2k) + a q^(4k) - a q^(n+2k) +
       a^2/(x y) q^(n+2k)) /
     (qPochhammer[q^2,q^2,k] qPochhammer[a,q^4,k] qPochhammer[a q^2/b,q^2,k] *
      qPochhammer[a q/x,q,2k] qPochhammer[a q/y,q,2k] qPochhammer[a q^(1+n),q,2k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a q,q,n] qPochhammer[a/(x y),q,n] qPochhammer[a q/b,q^2,k] qPochhammer[x,q,k] *
      qPochhammer[y,q,k] qPochhammer[q^(-n),q,k] q^k /
     (qPochhammer[a q/x,q,n-1] qPochhammer[a q/y,q,n-1] qPochhammer[a q,q^2,k] *
      qPochhammer[a q/b,q,k] qPochhammer[x y/a q^(1-n),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

(2.4) (corrected)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k qPochhammer[c,q,n] *
      qPochhammer[c/(a b q),q,n] /
     (qPochhammer[c,q,k] qPochhammer[a b/c q^(2-n),q,k] qPochhammer[q,q,k] *
      qPochhammer[c/a,q,n-1] qPochhammer[c/b,q,n-1] (1 + c/q - c/(a q) - c/(b q) -
      c q^(n-1) + c^2 q^(n-2)/(a b))), {k, 0, n}, n, 1]

(2.6) with x^2->x

qZeil[qPochhammer[a,q^3,2k] qPochhammer[a q^12,q^12,k] qPochhammer[x,q^2,3k] *
      qPochhammer[q^(-2n),q^2,3k] (a^2 q^(6n+3)/x^3)^k /
     (qPochhammer[q^3,q^3,2k] qPochhammer[a,q^12,k] qPochhammer[a q^2/x,q^2,3k] *
      qPochhammer[a q^(2n+2),q^2,3k]), {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a q^2,q^2,n] qPochhammer[q/x,q^2,n] qPochhammer[a,q^3,k] qPochhammer[x,q,k] *
      qPochhammer[q^(-2n),q,k] q^k /
     (qPochhammer[q,q^2,n] qPochhammer[a q^2/x,q^2,n] qPochhammer[a,q^2,k] *
      qPochhammer[x q^(1-2n),q^2,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 3]

(2.7) with x^2->x, a->a q^(-n) (LHS with k->2n-k)

qZeil[qPochhammer[q^(-3n),q,2k] q^k / (qPochhammer[q^(1-3n),q,k] qPochhammer[a,q,k] *
      qPochhammer[q^(1-3n)/a,q,k] qPochhammer[q,q,k]) /. {k->2n-k, a->a q^(-n)},
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[a^3,q^3,2n] qPochhammer[q^(-3n),q^3,2k] q^(3k) /
     (qPochhammer[a,q,3n] qPochhammer[a^3,q^3,n] qPochhammer[q^(3-3n),q^3,k] *
      qPochhammer[a^3 q^(3n),q^3,k] qPochhammer[q^(3-6n)/a^3,q^3,k] qPochhammer[q^3,q^3,k]) /. a->a q^(-n),
      {k, -Infinity, Infinity}, n, 2]

Proof of (2.7)

qZeil[qPochhammer[q^(-n),q,3k] q^(3k) qPochhammer[q^(-3m+1),q,n] qPochhammer[q^(-3m),q^3,n] /
     (qPochhammer[q^(-3m+3),q^3,k] qPochhammer[q^(3-3n+3m),q^3,k] qPochhammer[q^3,q^3,k] *
      qPochhammer[q^(-3m),q,2n]), {k, 0, n}, n, 1, {m}]

(2.8)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[x,q^2,k] qPochhammer[y,q^2,k] *
      qPochhammer[q^(-n),q,k] (a^2 q^(2+n)/(x y))^k /
     (qPochhammer[a,q^2,k] qPochhammer[a^2 q^2/x,q^2,k] qPochhammer[a^2 q^2/y,q^2,k] *
      qPochhammer[a q^(1+n),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[qPochhammer[a^2 q^2,q^2,n] qPochhammer[a^2 q^2/(x y),q^2,n] qPochhammer[x,q^2,k] *
      qPochhammer[y,q^2,k] qPochhammer[q^(-n),q,2k] q^(2k) /
     (qPochhammer[a^2 q^2/x,q^2,n] qPochhammer[a^2 q^2/y,q^2,n] qPochhammer[-a q,q,2k] *
      qPochhammer[x y/a^2 q^(-2n),q^2,k] qPochhammer[q^2,q^2,k]),
      {k, -Infinity, Infinity}, n, 2]

(2.13) with a->a q^(-n) (LHS with k->n-k)

qZeil[qPochhammer[a^2 q^2,q^2,3n-k] a^(2k) q^(2k^2) /
     (qPochhammer[q^2,q^2,k] qPochhammer[-a q,q,2k] qPochhammer[q^6,q^6,n-k]) /. {k->n-k, a->a q^(-n)},
      {k, -Infinity, Infinity}, n, 3]

qZeil[(1+a^3 q^(6n+3)) qPochhammer[a^6 q^12,q^6,2n-1] qPochhammer[a q,q,k] *
      (1-a^3 q^(6k+3)) (-1)^k a^(3k) q^(k(7k+1)/2) /
     (qPochhammer[q,q,k] qPochhammer[q^6,q^6,n-k] qPochhammer[a^6 q^12,q^6,n+k]) /. a->a q^(-n),
      {k, -Infinity, Infinity}, n, 3]

(2.14) with a->a q^(-n) (LHS with k->n-k)

qZeil[qPochhammer[a^2 q^2,q^2,3n-k] (-1)^k a^(2k) q^(3k^2) /
     (qPochhammer[q^2,q^2,k] qPochhammer[-a q,q,2k] qPochhammer[q^6,q^6,n-k]) /. {k->n-k, a->a q^(-n)},
      {k, -Infinity, Infinity}, n, 3]

qZeil[(1+a^3 q^(6n+3)) qPochhammer[a^6 q^12,q^6,2n-1] qPochhammer[a q,q,k] *
      (1-a^3 q^(6k+3)) (-1)^k a^(2k) q^(k(5k-1)/2) /
     (qPochhammer[q,q,k] qPochhammer[q^6,q^6,n-k] qPochhammer[a^6 q^12,q^6,n+k]) /. a->a q^(-n),
      {k, -Infinity, Infinity}, n, 3]

(2.22)

qZeil[qPochhammer[a^2,q^2,k] (1-a^2 q^(4k)) (-1)^k a^(6k) q^(k(7k-1)) /
     (qPochhammer[q^2,q^2,k] (1-a^2) qPochhammer[a^2 q,q,n+2k] qPochhammer[q,q,n-2k]),
      {k, 0, Infinity}, n, 3]

qZeil[a^(2k) q^(k^2) / (qPochhammer[q,q,k] qPochhammer[q,q,n-k] qPochhammer[a^2 q,q^2,k]),
      {k, 0, n}, n, 3]

(2.23)

qZeil[qPochhammer[a^2,q^2,k] (1-a^2 q^(4k)) (-1)^k a^(4k) q^(k(5k-1)) /
     (qPochhammer[q^2,q^2,k] (1-a^2) qPochhammer[a^4 q^2,q^2,n+2k] qPochhammer[q^2,q^2,n-2k]),
      {k, 0, Infinity}, n, 2]

qZeil[a^(2k) q^(k(3k-1)/2) / (qPochhammer[-a^2 q,q,2n] qPochhammer[q,q,k] *
      qPochhammer[q^2,q^2,n-k] qPochhammer[a^2 q,q^2,k]), {k, 0, n}, n, 2]

(2.24)

qZeil[qPochhammer[a^2 q^2,q^2,k] (1-a^4 q^(8k+4)) (-1)^k a^(4k) q^(k(5k+1)) /
     (qPochhammer[q^2,q^2,k] qPochhammer[a^4 q^6,q^2,n+2k] qPochhammer[q^2,q^2,n-2k]),
      {k, -Infinity, Infinity}, n, 2]

qZeil[(1-a^4 q^2)(1-a^4 q^4) a^(2k) q^(k(3k-1)/2) /
     (qPochhammer[-a^2 q,q,2n] (1-a^4 q^(4n+2)) qPochhammer[q,q,k] qPochhammer[q^2,q^2,n-k] *
      qPochhammer[a^2 q,q^2,k]), {k, 0, n}, n, 2]

(2.28) with a^2->a q^(-2n) (LHS 440 sec, RHS with k->n-k)

qZeil[qPochhammer[a,q^2,k] (1-a q^(4k)) qPochhammer[b,q^2,k] (a^3/b)^k q^(6k^2) /
     (qPochhammer[q^2,q^2,k] (1-a) qPochhammer[a q^2/b,q^2,k] *
      qPochhammer[a^3 q^3,q^3,n+2k] qPochhammer[q^3,q^3,n-2k]) /. a->a q^(-2n),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[a q/b,q^2,k] qPochhammer[a q,q,3n-k] a^k q^(k^2) /
     (qPochhammer[a^3 q^3,q^3,2n] qPochhammer[q,q,k] qPochhammer[q^3,q^3,n-k] *
      qPochhammer[a q,q^2,k] qPochhammer[a q/b,q,k]) /. {a->a q^(-2n), k->n-k},
      {k, -Infinity, Infinity}, n, 3]

(5.1) with b^2->b, c^2->c, e^2->e

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[-a q^2,q^2,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[-e q^n,q,k] qPochhammer[q^(-n),q,k] (a^2 q^3/(b c e))^k /
     (qPochhammer[q^2,q^2,k] qPochhammer[-a,q^2,k] qPochhammer[a^2 q^2/b,q^2,k] *
      qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[-a q^(1+n),q,k] qPochhammer[a/e q^(1-n),q,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[-a q,q,n] qPochhammer[e,q,n] qPochhammer[a,q,2k] qPochhammer[e^2 q^(2n),q^2,k] *
      qPochhammer[a^2 q^2/(b c),q^2,k] qPochhammer[q^(-2n),q^2,k] q^(2k) /
     (a^n qPochhammer[-q,q,n] qPochhammer[e/a,q,n] qPochhammer[a^2 q^2/b,q^2,k] *
      qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[e,q,2k] qPochhammer[q^2,q^2,k]),
      {k, -Infinity, Infinity}, n, 3]

(5.2) with b^2->b, c^2->c, e^2->e

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[-e q^n,q,k] qPochhammer[q^(-n),q,k] (a^2 q^2/(b c e))^k /
     (qPochhammer[q^2,q^2,k] qPochhammer[a^2,q^4,k] qPochhammer[a^2 q^2/b,q^2,k] *
      qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[-a q^(1+n),q,k] qPochhammer[a/e q^(1-n),q,k]),
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[-a q,q,n] qPochhammer[e,q,n] qPochhammer[e q^2,q^2,n] qPochhammer[a q,q,2k] *
      qPochhammer[e^2 q^(2n),q^2,k] qPochhammer[a^2 q^2/(b c),q^2,k] *
      qPochhammer[q^(-2n),q^2,k] q^(2k) /
     ((a q)^n qPochhammer[-q,q,n] qPochhammer[e/a,q,n] qPochhammer[e,q^2,n] *
      qPochhammer[a^2 q^2/b,q^2,k] qPochhammer[a^2 q^2/c,q^2,k] qPochhammer[e q,q,2k] *
      qPochhammer[q^2,q^2,k]), {k, -Infinity, Infinity}, n, 3]

(5.3) (inversion trick)

qZeil[(-1)^k q^(2(Binomial[k,2]+n)) qPochhammer[e^2 q^2,q^2,k-1] *
      (1-e^2 q^(4k)) qPochhammer[-q,q,k] qPochhammer[e/a^2,q,k] a^(2k) *
      qPochhammer[a^4 q^2,q^2,n] qPochhammer[e,q,2n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,n-k] qPochhammer[e^2 q^2,q^2,n+k] *
      qPochhammer[e,q,k] qPochhammer[-a^2 q,q,k] qPochhammer[a^2,q,2n] q^(2n)),
      {k, 0, n}, n, 1]

(5.4) (inversion trick)

qZeil[(-1)^k q^(2(Binomial[k,2]+n)) qPochhammer[e^2 q^2,q^2,k-1] *
      (1-e^2 q^(4k)) qPochhammer[-q,q,k] qPochhammer[e,q^2,k] qPochhammer[e q/a^2,q,k] *
      a^(2k) qPochhammer[a^4,q^2,n] qPochhammer[e q,q,2n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[q^2,q^2,n-k] qPochhammer[e^2 q^2,q^2,n+k] *
      qPochhammer[e,q,k] qPochhammer[-a^2,q,k] qPochhammer[e q^2,q^2,k] qPochhammer[a^2,q,2n] q^(2n)),
      {k, 0, n}, n, 1]

(5.8)

qZeil[qPochhammer[a^2,q^2,k] qPochhammer[a^2 q^4,q^4,k] qPochhammer[-e q^n,q,k] *
      qPochhammer[q^(-n),q,k] (1/e)^k (a q)^n qPochhammer[-q,q,n] qPochhammer[e/a,q,n] *
      qPochhammer[e,q^2,n] /
     (qPochhammer[q^2,q^2,k] qPochhammer[a^2,q^4,k] qPochhammer[-a q^(1+n),q,k] *
      qPochhammer[a/e q^(1-n),q,k] qPochhammer[-a q,q,n] qPochhammer[e,q,n] qPochhammer[e q^2,q^2,n]),
      {k, 0, n}, n, 1]

(v), p.267 with b^2->b, c^2->c, k->n-k (corrected, LHS 190 sec)

qZeil[qPochhammer[q^(-2n),q^2,k] (1+q^(2k-n)) qPochhammer[b,q^2,k] qPochhammer[c,q^2,k] *
      qPochhammer[-x,q,k] (-q^(3-2n)/(x b c))^k q^Binomial[k,2] /
     ((1+q^(-n)) qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^(2-2n)/c,q^2,k] *
      qPochhammer[q^2,q^2,k] qPochhammer[q^(1-n)/x,q,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[-1,q,n] (-x)^n qPochhammer[q^(-n),q,2k] qPochhammer[q^(2-2n)/(b c),q^2,k] *
      qPochhammer[x^2,q^2,k] (-q/x^2)^k q^(-2Binomial[k,2])/
     (qPochhammer[x,q,n] qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^(2-2n)/c,q^2,k] *
      qPochhammer[q^2,q^2,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 3]

(vi), p.268 with b^2->b, c^2->c, k->n-k (corrected, LHS 430 sec)

qZeil[qPochhammer[q^(-2n),q^2,k] (1-q^(4k-2n)) qPochhammer[b,q^2,k] *
      qPochhammer[c,q^2,k] qPochhammer[-x,q,k] (-q^(2-2n)/(x b c))^k q^Binomial[k,2] /
     ((1-q^(-2n)) qPochhammer[q^(2-2n)/b,q^2,k] qPochhammer[q^(2-2n)/c,q^2,k] *
      qPochhammer[q^2,q^2,k] qPochhammer[q^(1-n)/x,q,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 3]

qZeil[qPochhammer[-1,q,n] qPochhammer[q^(-n+1),q,2k] qPochhammer[q^(2-2n)/(b c),q^2,k] *
      qPochhammer[x^2,q^2,k] (-1/(x^2 q))^k q^(-2Binomial[k,2])/
     ((-x)^(1-n) qPochhammer[x,q,n] qPochhammer[q^(2-2n)/b,q^2,k] *
      qPochhammer[q^(2-2n)/c,q^2,k] qPochhammer[q^2,q^2,k]) /. k->n-k,
      {k, -Infinity, Infinity}, n, 3]

Koornwinder [1997]

(6.11)

qZeil[(1-q^k)/(1-q^n) qBinomial[n,k,q] y^(n-k) qPochhammer[y,q,k],
      {n, k, Infinity}, k, 1]

Krattenthaler [1984]

(p. 343, inv. rel. (i), (iii)) with p->o

qZeil[(-1)^(n-k) qBinomial[n+o,n-k,q] q^Binomial[k-m,2] *
      qBinomial[k+o,k-m,q], {k, m, n}, n, 0, {o}]

qZeil[(-1)^(n-k) qBinomial[2n+o,n-k,q] q^Binomial[k-m,2] *
      qBinomial[k+o+m,k-m,q] qBrackets[2k+o,q] /
      qBrackets[k+o+m,q], {k, m, n}, n, 0, {o}]

Krattenthaler [1996]

(4.3)

pqTelescope[qPochhammer[b p^l q^l,p,k-l] qPochhammer[b p^l/q^l/a,p,k-l] p^(k-n) *
            (1-b p^k q^k) (1-a q^k/p^k/b) qPochhammer[b p^n q^n,1/p,n-k] *
            qPochhammer[b p^n/q^n/a,1/p,n-k] /
           (qPochhammer[a q^(2l+1),q,k-l] qPochhammer[q,q,k-l] (1-b p^n q^n) *
            (1-a q^n/p^n/b) qPochhammer[a q^(2n-1),1/q,n-k] qPochhammer[1/q,1/q,n-k]),
            {k, l, n}]

Lakin [1952]

(27)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k qPochhammer[e,q,n] *
      qPochhammer[e/(a b),q,n] e^2 (1-a/e)(1-b/e)(1-q^(-n)/e) /
     (qPochhammer[a b q^(1-n)/e,q,k] qPochhammer[e q,q,k] qPochhammer[q,q,k] qPochhammer[e/a,q,n] *
      qPochhammer[e/b,q,n] (1-e)(a b+b q^(-n)+a q^(-n)-a b q^(-n)/e-e-
      a b q^(-n))), {k, 0, n}, n, 1]

(28)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^(2k) qPochhammer[e,q,n] *
      qPochhammer[e/(a b),q,n] e^2 (1-a/e)(1-b/e)(1-q^(-n)/e) /
     (qPochhammer[a b q^(1-n)/e,q,k] qPochhammer[e q,q,k] qPochhammer[q,q,k] qPochhammer[e/a,q,n] *
      qPochhammer[e/b,q,n] (1-e)(a+b+q^(-n)-1-a b q^(-n)/e-e)),
      {k, 0, n}, n, 1]

(29)

qZeil[qPochhammer[a,q,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] qPochhammer[l q,q,k] q^k *
      qPochhammer[e,q,n] qPochhammer[e/(a b),q,n] (1-l) e (1-e/l)(1-a/e)(1-b/e) *
      (1-q^(-n)/e) /
     (qPochhammer[a b q^(1-n)/e,q,k] qPochhammer[e q,q,k] qPochhammer[l,q,k] qPochhammer[q,q,k] *
      qPochhammer[e/a,q,n] qPochhammer[e/b,q,n] (1-l/e)(1-e)l((1-1/l)*
      (1-a b q^(-n)/(e l))(1-e/l)-(1-a/l)(1-b/l)(1-q^(-n)/l))),
      {k, 0, n}, n, 1]

Paule [1982]

3.4

qZeil[(-1)^k qBinomial[n,k,q] qBinomial[o-k,m-k,q] q^(-m n) *
      q^(1/2 k(k-1)) / qBinomial[o-n,m,q],
      {k, 0, n}, n, 1, {m, o}]

3.7

qZeil[qBinomial[n,k,q]^2 qBinomial[a+k,2n,q] q^((n-k)(n-k)) /
      qBinomial[a,n,q]^2, {k, 0, n}, n, 1, {a}]

3.8

qZeil[qBinomial[n,k,q]^2 qBinomial[u+2n-k,2n,q] q^(k^2) /
      qBinomial[u+n,n,q]^2, {k, 0, n}, n, 1, {u}]

3.9

qZeil[qBinomial[n,k,q] qBinomial[w,n-k,q] qBinomial[u+w+k,n+w,q] *
      q^((n-k)(n-k)) /
     (qBinomial[u+n,n,q] qBinomial[u+w,w,q]), {k, 0, n}, n, 1, {u, w}]

3.10

qZeil[qBinomial[x+y+k,k,q] qBinomial[x+n-b,n-k,q] *
      qBinomial[y+b-n,b-k,q] q^((n-k)(b-k)) /
     (qBinomial[x+n,n,q] qBinomial[y+b,b,q]),
      {k, 0, n}, n, 1, {b, x, y}]

Paule [1985a]

(V2)

qZeil[qBinomial[r,k,q] qBinomial[s,n-k,q] q^(k(s-n+k)) /
      qBinomial[r+s,n,q], {k, 0, n}, n, 1, {r, s}]

(10)

qZeil[qBinomial[2n,k+n,q] (-1)^k q^(k(3k-1)/2) qPochhammer[q,q,n] /
      qPochhammer[q,q,2n], {k, -n, n}, n, 1, MagicFactor->-k]

(11)

qZeil[qBinomial[2n+1,k+n,q] (-1)^k q^(k(3k-1)/2) qPochhammer[q,q,n] /
      qPochhammer[q,q,2n+1], {k, -n, n+1}, n, 1, MagicFactor->-k+1]

(16)

qZeil[(-1)^k qBinomial[2n,n+k,q^2] q^(k(3k-1)/2) / qPochhammer[q,q^2,n],
      {k, -n, n}, n, 1, MagicFactor->-k]

(17)

qZeil[(-1)^k qBinomial[2n+1,n+k,q^2] q^(k(3k-1)/2) / qPochhammer[q,q^2,n+1],
      {k, -n, n+1}, n, 1, MagicFactor->-k+1]

(18)

qZeil[(-1)^k qBinomial[2n+1,n+k,q^2] q^(k(3k+1)/2) /
     (qPochhammer[q,q^2,n+1] (1+q^(2n+1))), {k, -n, n+1}, n, 1, MagicFactor->-k+1]

(21)

qZeil[(-1)^k qBinomial[2n,n+k,q^2] q^(k(2k-1)) /
     (qPochhammer[-q^2,q^2,n] qPochhammer[q,q^2,n]), {k, -n, n}, n, 1, MagicFactor->-k]

(22)

qZeil[(-1)^k qBinomial[2n+1,n+k,q^2] q^(k(2k-1)) /
     (qPochhammer[-q^2,q^2,n] qPochhammer[q,q^2,n+1]),
      {k, -n, n+1}, n, 1, MagicFactor->-k+1]

(38)

qZeil[(-1)^k qBinomial[2n,n+k,q] q^((5k^2-3k)/2),
      {k, -n, n}, n, 2, MagicFactor->-k]

qZeil[(-1)^k qBinomial[n,k,q] q^(k^2+k) qPochhammer[q^(n+1),q,n], {k, 0, n}, n, 2]

(39)

qZeil[(-1)^k qBinomial[2n,n+k,q] q^((5k^2-k)/2),
      {k, -n, n}, n, 2, MagicFactor->-k]

qZeil[qBinomial[n,k,q] q^(k^2) qPochhammer[q^(n+1),q,n], {k, 0, n}, n, 2]

(40)

qZeil[(-1)^k q^(k^2) qBinomial[2n,n+k,q] / qPochhammer[q,q^2,n], {k, -n, n}, n, 1]

(41)

qZeil[(-1)^k qBinomial[2n,k,q] / qPochhammer[q,q^2,n], {k, 0, 2n}, n, 1]

(42)

qZeil[(-1)^k q^(2k^2-k) qBinomial[2n,n+k,q],
      {k, -n, n}, n, 2, MagicFactor->-k]

qZeil[qPochhammer[q^(n+1),q,n] q^(k^2+k) qBinomial[n,k,q] / qPochhammer[-q,q,k],
      {k, 0, n}, n, 2]

(43)

qZeil[(-1)^k qBinomial[2n,n+k,q] q^(2k^2), {k, -n, n}, n, 2]

qZeil[qBinomial[n,k,q] q^(k^2) qPochhammer[q^(n+1),q,n] / qPochhammer[-q,q,k],
      {k, 0, n}, n, 2]

(47)

qZeil[(-1)^k qBinomial[2n,n+k,q^2] q^((7k^2-3k)/2),
      {k, -n, n}, n, 3, MagicFactor->-k]

qZeil[qBinomial[n,k,q^2] q^(2k^2+2k) qPochhammer[q^(2n+2),q^2,n] /
      qPochhammer[-q,q,2k], {k, 0, n}, n, 3]

(48)

qZeil[(-1)^k qBinomial[2n,n+k,q^2] q^((7k^2-k)/2),
      {k, -n, n}, n, 3, MagicFactor->-k]

qZeil[qBinomial[n,k,q^2] q^(2k^2) qPochhammer[q^(2n+2),q^2,n] / qPochhammer[-q,q,2k],
      {k, 0, n}, n, 3]

(49)

qZeil[(-1)^k qBinomial[2n+1,n+k,q^2] q^((7k^2-5k)/2),
      {k, -n, n+1}, n, 3, MagicFactor->-k+1]

qZeil[qBinomial[n,k,q^2] q^(2k^2+2k) qPochhammer[q^(2n+2),q^2,n+1] /
      qPochhammer[-q,q,2k+1], {k, 0, n}, n, 3]

(55)

qZeil[(-1)^k qBinomial[2n,n+k,q^2] q^(4k^2-k),
      {k, -n, n}, n, 2, MagicFactor->-k]

qZeil[qBinomial[n,k,q^2] q^(2k^2) qPochhammer[q^(2n+2),q^2,n] / qPochhammer[-q,q^2,k],
      {k, 0, n}, n, 2]

(56)

qZeil[(-1)^k qBinomial[2n+1,n+k,q^2] q^(4k^2-3k),
      {k, -n, n+1}, n, 2, MagicFactor->-k+1]

qZeil[qBinomial[n,k,q^2] q^(2k^2+2k) qPochhammer[q^(2n+2),q^2,n+1] /
      qPochhammer[-q,q^2,k+1], {k, 0, n}, n, 2]

(60)

qZeil[(-1)^k qBinomial[2n,n+k,q^2] q^((k^2-k)/2) /
     ((-1)^n q^(n^2) qPochhammer[q,q^2,n]), {k, -n, n}, n, 1, MagicFactor->-k]

Paule [1985b]

(14)

qZeil[qPochhammer[q,q,a+b] q^(k^2-j^2) /
     (qPochhammer[q,q,a-k] qPochhammer[q,q,k+j] qPochhammer[q,q,k-j] qPochhammer[q,q,b-k] *
      qBinomial[a+b,a-j,q] qBinomial[a+b,a+j,q]),
      {k, -Infinity, Infinity}, j, 1, {a, b}]

(22) with p->o

qZeil[qBinomial[o+n-k,n-k,q] (-1)^(k-m) q^Binomial[k-m,2] *
      qBinomial[o+1,k-m,q], {k, m, n}, n, 0, {o}]

Paule [1987]

last identity of Ch. 2

qZeil[(-1)^k q^(k^2) qBinomial[2n+1,n-k,q] / qPochhammer[q,q^2,n+1],
      {k, -n-1, n}, n, 1, MagicFactor->-k-1]

(32) with u/v->x, rho:=0 (corrected)

qZeil[qBinomial[b+c,c+k,q] qBinomial[c+a,a+k,q] *
      qBinomial[a+b,b+k,q] q^(k(3k-1)/2) x^k,
      {k, -Infinity, Infinity}, a, 3, {b, c}]

qZeil[qFactorial[a+b+c-k,q] qPochhammer[-q/x,q,k] qPochhammer[-x,q,k] q^(k^2) /
     (qFactorial[a-k,q] qFactorial[b-k,q] qFactorial[c-k,q] *
      qFactorial[2k,q]), {k, -Infinity, Infinity}, a, 3, {b, c}]

(32) with u/v->x, rho:=1 (corrected)

qZeil[qBinomial[b+c+1,c+k,q] qBinomial[c+a+1,a+k,q] *
      qBinomial[a+b+1,b+k,q] q^(k(3k-1)/2) x^k,
      {k, -Infinity, Infinity}, a, 3, {b, c}]

qZeil[qFactorial[a+b+c-k+1,q] qPochhammer[-1/x,q,k] qPochhammer[-x q,q,k+1] q^(k^2+k) /
     (qFactorial[a-k,q] qFactorial[b-k,q] qFactorial[c-k,q] *
      qFactorial[2k+1,q]), {k, -Infinity, Infinity}, a, 3, {b, c}]

Paule [1994]

(8)

qZeil[q^(k^2) / (qPochhammer[q,q,k] qPochhammer[q,q,n-k]), {k, 0, n}, n, 2]

qZeil[(-1)^k q^((5k^2-k)/2) / (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]),
      {k, -n, n}, n, 2, MagicFactor->-k]

(13) with a:=0

qZeil[qBinomial[2n-k,k,q] q^(k^2), {k, 0, n}, n, 2]

qZeil[qBinomial[2n+2,n-5k,q] q^(10k^2-k) qBrackets[10k+2,q] /
      qBrackets[2n+2,q], {k, -Infinity, Infinity}, n, 2]

(13) with a:=1

qZeil[qBinomial[2n-k+1,k,q] q^(k^2+k), {k, 0, n}, n, 2]

qZeil[qBinomial[2n+4,n-5k,q] q^(10k^2+3k) qBrackets[10k+4,q] /
      qBrackets[2n+4,q], {k, -Infinity, Infinity}, n, 2]

(21)

qZeil[qBinomial[2n,n+k,q] q^Binomial[k,2] x^k /
     (qPochhammer[-q/x,q,n] qPochhammer[-x,q,n]), {k, -n, n}, n, 1]

(27)

qZeil[(-1)^k qBinomial[2n,n+k,q] q^((3k^2+k)/2) (w^3)^k *
      (1-w^2 q^(2k+1)) qPochhammer[1/w,q,n] qPochhammer[w q,q,n] /
     (qPochhammer[1/w^2,q,n-k] qPochhammer[w^2 q^2,q,n+k] (1-w^2 q)),
      {k, -n, n}, n, 1]

Paule, Riese [1997]

(14)

qZeil[(-1)^k q^(4k^2) qBinomial[2n,n-4k,q],
      {k, -Infinity, Infinity}, n, 3]

qZeil[q^(2k^2) qBinomial[n,2k,q^2] qPochhammer[-q,q^2,n-2k] qPochhammer[-1,q^4,k],
      {k, 0, Infinity}, n, 3]

Paule, Strehl [1995]

Example from Section 6

qTelescope[(-1)^k q^(Binomial[k,2]-(n-1)k) qBinomial[n,k,q],
           {k, 0, m}, {n}]

(37)

qZeil[q^(6k^2+k) qBinomial[2n+2,n-3k,q] (1-q^(6k+2)) / (1-q^(2n+2)),
      {k, -n, n}, n, 1]

Rahman, Verma [1993]

(3.1)

qZeil[qPochhammer[q^a,q,k] qPochhammer[q^b,q,k] (x q^(1-2b))^k /
     (qPochhammer[q^(1+a-b),q,k] qPochhammer[q,q,k]), {k, 0, Infinity}, a, 2, {b}]

qZeil[qPochhammer[x q^(1-b),q,a] qPochhammer[x q^(a-b),q,k] (1-x q^(a-b+2k)) *
      qPochhammer[q^a,q,2k] qPochhammer[x,q,k] (x q^(1-2b))^k /
     (qPochhammer[x^2 q^(1-2b),q,a] (1-x q^(a-b)) qPochhammer[x^2 q^(1+a-2b),q,2k] *
      qPochhammer[q^(1+a-b),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, a, 2, {b}]

(3.13), (3.15) (1st RHS corrected)

qZeil[qPochhammer[x q^(b-1),q,k] (1-x q^(b-1+2k)) qPochhammer[q^(2a),q,2k] *
      qPochhammer[x,q,k] (x q^(2b-4a-1))^k /
     ((1-x q^(b-1)) qPochhammer[x^2 q^(2b-2a-1),q,2k] qPochhammer[q^b,q,k] *
      qPochhammer[q,q,k]), {k, -Infinity, Infinity}, b, 2, {a}]

qZeil[qPochhammer[x^2 q^(2b-4a-1),q,2a] qPochhammer[q^(2a),q,k] *
      qPochhammer[q^(2a-b+1),q,k] (x q^(2b-4a-1))^k /
     (qPochhammer[x q^(b-2a),q,2a] qPochhammer[q^b,q,k] qPochhammer[q,q,k]),
      {k, 0, Infinity}, b, 2, {a}]

qZeil[qPochhammer[x^2 q^(2b-4a-1),q,2a] qPochhammer[q^(2a),q,k] *
      qPochhammer[q^(2b-2a-1),q,k] (x q^(b-2a))^k (-1)^k q^Binomial[k,2] /
     (qPochhammer[x q^(b-2a),q,2a] qPochhammer[x q^(2b-4a-1),q,2a] qPochhammer[q^b,q,k] *
      qPochhammer[x q^(2b-2a-1),q,k] qPochhammer[q,q,k]),
      {k, -Infinity, Infinity}, b, 2, {a}]

(3.27) (corrected)

qZeil[qPochhammer[x q^(c-1),q,k] (1-x q^(c-1+2k)) qPochhammer[q^(c-a),q,2k] *
      qPochhammer[x,q,k] (x q^(2a-1))^k /
     ((1-x q^(c-1)) qPochhammer[x^2 q^(c+a-1),q,2k] qPochhammer[q^c,q,k] *
      qPochhammer[q,q,k]), {k, -Infinity, Infinity}, a, 2, {c}]

qZeil[qPochhammer[x q^c,q,2a-c-1] qPochhammer[q^a,q,k] qPochhammer[q^(1-a),q,k] *
      (x q^(a+c-1))^k (-1)^k q^Binomial[k,2] /
     (qPochhammer[x^2 q^(a+c-1),q,a-c] qPochhammer[q^c,q,k] qPochhammer[x q^a,q,k] *
      qPochhammer[q,q,k]), {k, 0, Infinity}, a, 2, {c}]

(4.5) with x->x q^a (corrected)

qZeil[qPochhammer[-x q^(3b-a-1),q,k] (1+x q^(3b-a-1+2k)) qPochhammer[q^a,q,k] *
      qPochhammer[q^(2b),q^2,k] qPochhammer[-q^(2b-a),q,k] qPochhammer[-x q^(b-a),q,k] *
      (-x q^(b-a))^k /
     ((1+x q^(3b-a-1)) qPochhammer[-x q^(3b-2a),q,k] qPochhammer[x^2 q^(4b-2a),q^2,k] *
      qPochhammer[x q^b,q,k] qPochhammer[q^(2b),q,k] qPochhammer[q,q,k]) /. x->x q^a,
      {k, -Infinity, Infinity}, a, 2, {b}]

qZeil[qPochhammer[x^2 q^(2b-2a),q^2,a] qPochhammer[q^(2a),q^2,k] qPochhammer[q^(4b-2a),q^2,k] *
      (-x^2 q^(1+2b-2a))^k q^(2Binomial[k,2]) /
     (qPochhammer[-x q^(b-a),q,a] qPochhammer[-x q^(3b-2a),q,a] qPochhammer[q^(2b+1),q^2,k] *
      qPochhammer[x^2 q^(4b-2a),q^2,k] qPochhammer[q^2,q^2,k]) /. x->x q^a,
      {k, -Infinity, Infinity}, a, 2, {b}]

Slater [1951]

(3.1)

qZeil[(1-a q^(6k)) qPochhammer[q^(-n),q,3k] (a^2)^k q^(3n k) qPochhammer[e,q^3,k] *
      qPochhammer[a,q,2n] qPochhammer[a q/e,q,n] /
     ((1-a) qPochhammer[a q^(1+n),q,3k] e^k qPochhammer[q^3 a/e,q^3,k] qPochhammer[q,q,n] *
      qPochhammer[a q,q,n] qPochhammer[a^2/e,q^3,n]) /. a -> q^m,
      {k, -Infinity, Infinity}, n, 1, {m}]

(3.2) (corrected)

qZeil[(1-q^(6k+1)) (-1)^k qPochhammer[e,q^3,k] q^(k(9k+1)/2) qPochhammer[q,q,2n] *
      qPochhammer[q^2/e,q,n] /
     (qPochhammer[q,q,n+3k+1] qPochhammer[q,q,n-3k] qPochhammer[q^4/e,q^3,k] e^k *
      qPochhammer[q^2/e,q^3,n]), {k, -n, n}, n, 1]

(3.3)

qZeil[(1-q^(6k+1)) q^(6k^2-k) / (qPochhammer[q,q,n+3k+1] qPochhammer[q,q,n-3k]) *
      qPochhammer[q,q,2n], {k, -n, n}, n, 1]

(3.4)

qZeil[(1-q^(6k+1)) q^(3k^2-2k) / (qPochhammer[q,q,n+3k+1] qPochhammer[q,q,n-3k]) *
      qPochhammer[q,q,2n] / q^(n^2-n), {k, -n, n}, n, 1]

(3.5)

qZeil[(-1)^k (1-q^(6k+2)) q^((9k^2+5k)/2) qPochhammer[e,q^3,k] qPochhammer[q^2,q,2n] *
      qPochhammer[q^3/e,q,n] /
     (qPochhammer[q^2,q,n+3k+1] qPochhammer[q,q,n-3k] e^k qPochhammer[q^5/e,q^3,k] *
      qPochhammer[q^4/e,q^3,n]), {k, -n, n}, n, 1]

(3.6)

qZeil[(1-q^(6k+2)) q^(6k^2+k) / (qPochhammer[q^2,q,n+3k+1] qPochhammer[q,q,n-3k]) *
      qPochhammer[q^2,q,2n], {k, -n, n}, n, 1]

(3.7)

qZeil[(1-q^(6k+2)) q^(3k^2-k) / (qPochhammer[q^2,q,n+3k+1] qPochhammer[q,q,n-3k]) *
      qPochhammer[q^2,q,2n] / q^(n^2), {k, -n, n}, n, 1]

(3.8) (rewritten)

qZeil[(-1)^k (1+q^(3k)) q^(3k(3k-1)/2) qPochhammer[q,q,n] qPochhammer[q,q,2n-1] /
     (qPochhammer[q,q,n+3k] qPochhammer[q,q,n-3k] qPochhammer[q^3,q^3,n-1]),
      {k, -n, n}, n, 1]

(3.9)

qZeil[(1-q^(6k+1)) (-1)^k q^((9k^2-3k)/2) qPochhammer[q,q,2n] qPochhammer[q,q,n-1] /
     (qPochhammer[q,q,n+3k+1] qPochhammer[q,q,n-3k] qPochhammer[q^3,q^3,n-1]),
     {k, -n, n}, n, 1]

(3.10) with q->q^2

qZeil[(1-q^(12k+4)) (-1)^k q^(9k^2) qPochhammer[q^4,q^2,2n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^4,q^2,n+3k+1] qPochhammer[q^2,q^2,n-3k] qPochhammer[q^3,q^6,n]),
     {k, -n, n}, n, 1]

(4.1)

qZeil[(1-a q^(2k)) (-1)^k q^((k^2+k)/2) qPochhammer[a,q,k] qPochhammer[c,q,k] *
      qPochhammer[d,q,k] a^k qPochhammer[q,q,n] qPochhammer[a q/c,q,n] qPochhammer[a q/d,q,n] /
     (qPochhammer[a,q,n+k+1] qPochhammer[q,q,n-k] qPochhammer[q,q,k] qPochhammer[a q/c,q,k] *
      qPochhammer[a q/d,q,k] c^k d^k qPochhammer[a q/(c d),q,n]),
      {k, 0, n}, n, 1]

A (1)

qZeil[q^(6k^2) (q^(-k)+q^k-q^(5k+1)(1-q^(n-3k))/(1-q^(n+3k+1))-
      q^(-5k+1)(1-q^(n+3k))/(1-q^(n-3k+1))) qPochhammer[q,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q,q,n+3k]), {k, -n, n}, n, 1]

A (2)

qZeil[q^(6k^2) (q^k-(q^(5k+1)+q^(7k+2))(1-q^(n-3k))/(1-q^(n+3k+2))+
      q^(-k)(1-q^(n+3k+1))/(1-q^(n-3k+1))) qPochhammer[q^2,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q^2,q,n+3k]), {k, -n, n}, n, 1]

A (3)

qZeil[q^(6k^2) (q^(-2k)+q^(2k)-q^(2k)(1-q^(n-3k))/(1-q^(n+3k+1))-
      q^(-2k)(1-q^(n+3k))/(1-q^(n-3k+1))) qPochhammer[q,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q,q,n+3k] q^n), {k, -n, n}, n, 1]

A (4)

qZeil[q^(6k^2) (q^(4k)-(q^(8k+2)+q^(4k))(1-q^(n-3k))/(1-q^(n+3k+2))+
      q^(-4k)(1-q^(n+3k+1))/(1-q^(n-3k+1))) qPochhammer[q^2,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q^2,q,n+3k] q^n), {k, -n, n}, n, 1]

A (5)

qZeil[q^(3k^2) (q^(-k)+q^k-q^k(1-q^(n-3k))/(1-q^(n+3k+1))-
      q^(-k)(1-q^(n+3k))/(1-q^(n-3k+1))) qPochhammer[q,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q,q,n+3k] q^(n^2)), {k, -n, n}, n, 1]

A (6)

qZeil[q^(3k^2) (q^(-k)-(q^k+q^(5k+2))(1-q^(n-3k))/(1-q^(n+3k+2))+
      q^k(1-q^(n+3k+1))/(1-q^(n-3k+1))) qPochhammer[q^2,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q^2,q,n+3k] q^(n^2)), {k, -n, n}, n, 1]

A (7)

qZeil[q^(3k^2) (q^(-2k)+q^(2k)-q^(4k+1)(1-q^(n-3k))/(1-q^(n+3k+1))-
      q^(-4k+1)(1-q^(n+3k))/(1-q^(n-3k+1))) qPochhammer[q,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q,q,n+3k] q^(n^2-n)), {k, -n, n}, n, 1]

A (8)

qZeil[q^(3k^2) (q^(2k)-(q^(4k+1)+q^(2k))(1-q^(n-3k))/(1-q^(n+3k+2))+
      q^(-2k)(1-q^(n+3k+1))/(1-q^(n-3k+1))) qPochhammer[q^2,q,2n] /
     (qPochhammer[q,q,n-3k] qPochhammer[q^2,q,n+3k] q^(n^2+n)), {k, -n, n}, n, 1]

B (1)

qZeil[(-1)^k q^(k(3k-1)/2) (1+q^k) qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]), {k, -n, n}, n, 1]

B (2)

qZeil[(-1)^k q^(3k(k-1)/2) (1+q^(3k)) qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k] q^n), {k, -n, n}, n, 1]

B (3)

qZeil[(-1)^k q^(k(3k+1)/2) (1-q^(2k+1)) qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k] (1-q)), {k, -n-1, n}, n, 1]

B (4)

qZeil[(-1)^k q^(k(3k-1)/2) ((1-q^k)/q + q^(3k)(1-q^(k+1))) qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k] (1-q) q^n), {k, -n-1, n}, n, 1]

C (1)

qZeil[(-1)^k q^(3k^2) (q^k+q^(-k)) qPochhammer[q,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k]), {k, -n, n}, n, 1]

C (2)

qZeil[(-1)^k q^(3k^2) (q^k+q^(-k)-q^k(1-q^(4k+2))(1-q^(n-2k))/
      (1-q^(n+2k+1))) qPochhammer[q,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] q^n), {k, -n, n}, n, 1]

C (3)

qZeil[(-1)^k q^(3k^2+k) (1-q^(4k+2)(1-q^(n-2k))/(1-q^(n+2k+2))) *
      qPochhammer[q^3,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k]), {k, -n-1, n}, n, 1]

C (4)

qZeil[(-1)^k q^(3k^2+3k) (1-(1-q^(n-2k))/(1-q^(n+2k+2))) *
      qPochhammer[q^3,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k] q^n), {k, -n-1, n}, n, 1]

C (5)

qZeil[(-1)^k q^(k^2) (q^k+q^(-k)) qPochhammer[q,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] q^((n^2-n)/2)), {k, -n, n}, n, 1]

C (6)

qZeil[(-1)^k q^(k^2) (q^(-k)-q^(3k+2)(1-q^(n-2k))/(1-q^(n+2k+2))) *
      qPochhammer[q^3,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k] q^((n^2-n)/2)), {k, -n-1, n}, n, 1]

C (7)

qZeil[(-1)^k q^(k^2+k) (1-(1-q^(n-2k))/(1-q^(n+2k+2))) *
      qPochhammer[q^3,q^2,n] qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k] q^((n^2+n)/2)), {k, -n-1, n}, n, 1]

E (1)

qZeil[q^(4k^2) (2-(2q^(4k+1))(1-q^(n-2k))/(1-q^(n+2k+1))) *
      qPochhammer[q^2,q^2,n] / (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k]), {k, -n, n}, n, 1]

E (2)

qZeil[(2-2(1-q^(n-2k))/(1-q^(n+2k+1))) qPochhammer[q^2,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] (-1)^n), {k, -n, n}, n, 1]

E (3)

qZeil[(-1)^k q^(k^2) (1-q^(2k+1)) qPochhammer[q,q,n] qPochhammer[-q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k] (1-q)), {k, -n-1, n}, n, 1]

E (4)

qZeil[q^(4k^2) (q^(2k)+q^(-2k)+(-q^(2k)-q^(6k+2))(1-q^(n-2k))/
      (1-q^(n+2k+1))) qPochhammer[q^2,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] q^n), {k, -n, n}, n, 1]

E (5)

qZeil[(q^(2k)+q^(-2k)+(-q^(2k+1)-q^(-2k-1))(1-q^(n-2k))/
      (1-q^(n+2k+1))) q^n qPochhammer[q^2,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] (-1)^n), {k, -n, n}, n, 1]

E (6) (corrected)

qZeil[(-1)^k q^(k^2-k) (1+q^(2k)) qPochhammer[q,q,n] qPochhammer[-q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k] q^n), {k, -n, n}, n, 1]

E (7) (corrected)

qZeil[(-1)^k (q^(-k)-q^(k+1)) q^n qPochhammer[q,q,n] qPochhammer[-q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k] (1-q) (-1)^n), {k, -n-1, n}, n, 1]

F (1) with q->q^2

qZeil[q^(2k^2-k) (1+q^(2k)) qPochhammer[q,q^2,n] qPochhammer[q^2,q^2,n] /
      (qPochhammer[q^2,q^2,n-k] qPochhammer[q^2,q^2,n+k]), {k, -n, n}, n, 1]

F (2) with q->q^2

qZeil[q^(2k^2+k) (1+q^(2k+1)) qPochhammer[q^3,q^2,n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^4,q^2,n+k] (1+q)), {k, -n-1, n}, n, 1]

F (3) with q->q^2

qZeil[(q^k+q^(-k)) q^n qPochhammer[q,q^2,n] qPochhammer[q^2,q^2,n] /
      (qPochhammer[q^2,q^2,n-k] qPochhammer[q^2,q^2,n+k]), {k, -n, n}, n, 1]

F (4) with q->q^2

qZeil[q^(-k) (1+q^(2k+1)) q^n qPochhammer[q^2,q^2,n] qPochhammer[q^3,q^2,n] /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^4,q^2,n+k] (1+q)), {k, -n-1, n}, n, 1]

G (1) with q->q^2

qZeil[q^(6k^2) (q^k+q^(-k)+(-q^(7k+2)-q^(5k+1))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[-q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k]), {k, -n, n}, n, 1]

G (2) with q->q^2 (corrected)

qZeil[q^(6k^2+k) (1-q^(4k+1)+q^(10k+5)(1-q^(-4k-3))(1-q^(2n-4k))/
      (1-q^(2n+4k+4))) qPochhammer[q^4,q^4,n] qPochhammer[-q^3,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^4,q^2,n+2k] (1-q)), {k, -n-1, n}, n, 1]

G (3) with q->q^2

qZeil[q^(6k^2) (q^(3k)+q^(-3k)+(-q^(3k)-q^(9k+3))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[-q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k] q^(2n)), {k, -n, n}, n, 1]

G (4) with q->q^2

qZeil[q^(2k^2) (q^(3k)+q^(-3k)+(-q^(5k+2)-q^(-k-1))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[-q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k] (-1)^n q^(n^2-2n)),
      {k, -n, n}, n, 1]

qZeil[q^(2k^2) (q^k+q^(-k)+(-q^k-q^(3k+1))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[-q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k] (-1)^n q^(n^2)),
      {k, -n, n}, n, 1]

G (5) with q->q^2 (corrected)

qZeil[q^(2k^2-k) (1-q^(4k+1)+q^(6k+3)(1-q^(-4k-3))(1-q^(2n-4k))/
      (1-q^(2n+4k+4))) qPochhammer[q^4,q^4,n] qPochhammer[-q^3,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^4,q^2,n+2k] (1-q) (-1)^n q^(n^2)),
      {k, -n-1, n}, n, 1]

H (1) with q->q^2 (corrected)

qZeil[q^(k^2-k) (1+q^(2k)) qPochhammer[q^2,q^4,n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^2,q^2,n+k] 2 qPochhammer[-q^2,q^2,n-1]),
      {k, -n, n}, n, 1]

H (2) with q->q^2

qZeil[(-1)^k q^(2k^2-k) (1+q^(2k)) qPochhammer[-q,q^2,n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^2,q^2,n+k]), {k, -n, n}, n, 1]

H (3)

qZeil[(-1)^k q^(k(-k-3)/2) (1+q^(3k)) q^(n(n+3)/2) qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k] (-1)^n), {k, -n, n}, n, 1]

H (4)

qZeil[(-1)^k q^(k(-k-1)/2) (1+q^k) q^(n(n+1)/2) qPochhammer[q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k] (-1)^n), {k, -n, n}, n, 1]

H (5)

qZeil[(q^(-k)-q^k) / (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]), {k, 1, n}, n, 0]

H (7)

qZeil[2 (-1)^k qPochhammer[q,q,n] qPochhammer[-q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k] (-1)^n), {k, -n, n}, n, 1]

H (8)

qZeil[(-1)^k (q^k+q^(-k)) q^n qPochhammer[q,q,n] qPochhammer[-q,q,n] /
     (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k] (-1)^n), {k, -n, n}, n, 1]

H (9)

qZeil[q^(k^2) (q^k-q^(-k)) / (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]),
      {k, 1, n}, n, 0]

H (10)

qZeil[q^(k(k-1)/2) (1-q^k) / (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]),
      {k, 1, n}, n, 0]

H (11)

qZeil[q^(k(k+1)/2) (1-q^(-k)) / (qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]),
      {k, 0, n}, n, 0]

H (12)

qZeil[q^(-k) (1-q^(2k+1)) / (qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k]),
      {k, 1, n}, n, 0]

H (14)

qZeil[-q^(k^2-1) (1-q^(2k+1)) / (qPochhammer[q,q,n-k] qPochhammer[q^2,q,n+k]),
      {k, 1, n}, n, 0]

H (15) with q->q^2

qZeil[(-1)^k q^(k^2) (1+q^(2k+1)) / (qPochhammer[q^2,q^2,n-k] qPochhammer[q^4,q^2,n+k]),
      {k, 1, n}, n, 0]

H (16) with q->q^2

qZeil[(-1)^k q^(k^2) (q^(2k)+q^(-1)) /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^4,q^2,n+k]), {k, 1, n}, n, 0]

I (1) with q->q^2

qZeil[(-1)^k q^(2k^2) (q^(3k)+q^(-3k)+(q^(-k-1)-q^(5k+2))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k] q^(n^2-2n)), {k, -n, n}, n, 1]

I (2) with q->q^2 (corrected)

qZeil[(-1)^k q^(2k^2) (q^k+q^(-k)-(q^k-q^(3k+1))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k] q^(n^2)), {k, -n, n}, n, 1]

I (3) with q->q^2

qZeil[(-1)^k q^(6k^2) (q^k+q^(-k)-(q^(7k+2)-q^(5k+1))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k]), {k, -n, n}, n, 1]

I (4) with q->q^2

qZeil[(-1)^k q^(6k^2) (q^(3k)+q^(-3k)-(q^(3k)-q^(9k+3))(1-q^(2n-4k))/
      (1-q^(2n+4k+2))) qPochhammer[q^4,q^4,n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k] q^(2n)), {k, -n, n}, n, 1]

I (5)

qZeil[q^(2k^2-k) (1-q^(4k+1)(1-q^(n-2k))/(1-q^(n+2k+1))) /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k]), {k, 1, n}, n, 0]

I (6)

qZeil[q^(2k^2+k) (1-(1-q^(n-2k))/(1-q^(n+2k+1))) /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] q^n), {k, 1, n}, n, 0]

I (7)

qZeil[(-1)^k q^(2k^2) (q^k+q^(-k)+(q^k-q^(3k+1))(1-q^(n-2k))/
      (1-q^(n+2k+1))) qPochhammer[q^2,q^2,n] qPochhammer[q,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] qPochhammer[-1,q^2,n]), {k, -n, n}, n, 1]

I (8) (corrected)

qZeil[(-1)^k q^(2k^2) (q^k+q^(-k)+(q^(3k+1)-q^k)(1-q^(n-2k))/
      (1-q^(n+2k+1))) qPochhammer[q^2,q^2,n] qPochhammer[q,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] qPochhammer[-1,q^2,n] q^n), {k, -n, n}, n, 1]

I (9)

qZeil[q^(2k^2) (q^k+q^(-k)) qPochhammer[q,q^2,n] qPochhammer[q,q,n] qPochhammer[q,q,n-1] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] qPochhammer[q^2,q^2,n-1]), {k, -n, n}, n, 1]

I (10)

qZeil[q^(2k^2) (1-q^(4k+2)(1-q^(n-2k))/(1-q^(n+2k+2))) /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k]), {k, 1, n}, n, 0]

I (11)

qZeil[q^(2k^2+2k) (1-(1-q^(n-2k))/(1-q^(n+2k+2))) /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k]), {k, 1, n}, n, 0]

I (12)

qZeil[(-1)^k q^(2k^2) (1-q^(4k+2)(1-q^(n-2k))/(1-q^(n+2k+2))) *
      qPochhammer[q^3,q^2,n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k] qPochhammer[-q,q^2,n]), {k, -n-1, n}, n, 1]

I (13)

qZeil[(-1)^k q^(2k^2+2k) (1-(1-q^(n-2k))/(1-q^(n+2k+2))) *
      qPochhammer[q^3,q^2,n] qPochhammer[q^2,q^2,n] /
     (qPochhammer[q,q,n-2k] qPochhammer[q^2,q,n+2k] q^n qPochhammer[-q,q^2,n]),
      {k, -n-1, n}, n, 1]

K (1) (n >= 1 !!!)

qZeil[q^(8k^2) (q^(2k)+q^(-2k)-q^(6k+1)(1-q^(n-4k))/(1-q^(n+4k+1))-
      q^(-6k+1)(1-q^(n+4k))/(1-q^(n-4k+1))) qPochhammer[q,q,2n] /
     (qPochhammer[q,q,n-4k] qPochhammer[q,q,n+4k] qPochhammer[-q^2,q^2,n-1]) /. n->n+1,
      {k, -n-1, n+1}, n, 1]

K (2) (n >= 1 !!!)

qZeil[q^(8k^2) (q^(2k)+q^(-2k)-q^(2k)(1-q^(n-4k))/(1-q^(n+4k+1))-
      q^(-2k)(1-q^(n+4k))/(1-q^(n-4k+1))) qPochhammer[q,q,2n] /
     (qPochhammer[q,q,n-4k] qPochhammer[q,q,n+4k] qPochhammer[-q^2,q^2,n-1] q^n) /. n->n+1,
      {k, -n-1, n+1}, n, 1]

K (3)

qZeil[q^(8k^2) (1-q^(8k+2)(1-q^(n-4k))/(1-q^(n+4k+2))+
      (1-q^(n+4k+1))/(1-q^(n-4k+1))-q^(-8k+2)(1-q^(n+4k+1))(1-q^(n+4k))/
      (1-q^(n-4k+1))/(1-q^(n-4k+2))) qPochhammer[q^2,q,2n] /
     (qPochhammer[q,q,n-4k] qPochhammer[q^2,q,n+4k] qPochhammer[-q,q^2,n]), {k, -n, n}, n, 1]

K (4)

qZeil[q^(8k^2) (q^(4k)-q^(4k)(1-q^(n-4k))/(1-q^(n+4k+2))+q^(-4k)*
      (1-q^(n+4k+1))/(1-q^(n-4k+1))-q^(-4k)(1-q^(n+4k+1))(1-q^(n+4k))/
      (1-q^(n-4k+1))/(1-q^(n-4k+2))) qPochhammer[q^2,q,2n] /
     (qPochhammer[q,q,n-4k] qPochhammer[q^2,q,n+4k] qPochhammer[-q,q^2,n] q^n),
      {k, -n, n}, n, 1]

K (5)

qZeil[q^(8k^2) (q^(2k)-(q^(10k+3)+q^(6k+1))(1-q^(n-4k))/(1-q^(n+4k+3))+
      q^(-2k)(1-q^(n+4k+2))(1-q^(n+4k+1))/(1-q^(n-4k+1))/(1-q^(n-4k+2))) *
      qPochhammer[q^3,q,2n] / (qPochhammer[q,q,n-4k] qPochhammer[q^3,q,n+4k] qPochhammer[-q^2,q^2,n]),
      {k, -n, n}, n, 1]

K (6)

qZeil[q^(8k^2) (q^(6k)-(q^(6k)+q^(10k+2))(1-q^(n-4k))/(1-q^(n+4k+3))+
      q^(-6k)(1-q^(n+4k+2))(1-q^(n+4k+1))/(1-q^(n-4k+1))/(1-q^(n-4k+2))) *
      qPochhammer[q^3,q,2n] /
     (qPochhammer[q,q,n-4k] qPochhammer[q^3,q,n+4k] qPochhammer[-q^2,q^2,n] q^n),
      {k, -n, n}, n, 1]

Slater [1952]

H (18) with q->q^2

qZeil[(1+q^(2k+1)) (-1)^k q^(k^2) (1-q^(2n+1)) qPochhammer[q^2,q^2,n]^2 /
     ((1+q) qPochhammer[q^2,q^2,n-k] qPochhammer[q^4,q^2,n+k] (1-q)),
      {k, 0, n}, n, 0]

H (19) with q->q^2

qZeil[(1+q^(2k+1)) q^(k^2) qPochhammer[q^3,q^2,n] qPochhammer[q^4,q^4,n] /
     ((1+q) qPochhammer[q^2,q^2,n-k] qPochhammer[q^4,q^2,n+k] qPochhammer[-q,q^2,n]),
      {k, 0, n}, n, 1]

I (14)

qZeil[(1+q^(2k)) (-1)^k q^(k(2k-1)) qPochhammer[q,q^2,n] qPochhammer[q,q,n] *
      qPochhammer[-q,q,n-1] /
     (qPochhammer[q,q,n-2k] qPochhammer[q,q,n+2k] qPochhammer[-q^2,q^2,n-1]),
      {k, -n, n}, n, 1]

I (15) with q->q^2 (corrected)

qZeil[q^(4k^2) (1+q^(4k+1)+q^(8k+4)(1+q^(-4k-3))(1-q^(2n-4k))/
      (1-q^(2n+4k+4))) qPochhammer[q^4,q^4,n] qPochhammer[q,q^2,n] qPochhammer[q^3,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^4,q^2,n+2k] qPochhammer[q^2,q^4,n] (1+q)),
      {k, -n-1, n}, n, 1]

I (16) with q->q^2 (corrected)

qZeil[(-1)^k q^(4k^2) (1+q^(4k+1)-q^(8k+4)(1+q^(-4k-3))(1-q^(2n-4k))/
      (1-q^(2n+4k+4))) qPochhammer[q^4,q^4,n] qPochhammer[-q,q^2,n] qPochhammer[q^3,q^2,n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^4,q^2,n+2k] qPochhammer[-q^2,q^4,n] (1+q)),
      {k, -n-1, n}, n, 1]

I (17)

qZeil[(1+q^(2k+1)) q^(k(2k+1)) qPochhammer[q,q,n] qPochhammer[q^2,q,n] *
      qPochhammer[q^3,q^2,n] /
     ((1+q) qPochhammer[q,q,n-2k] qPochhammer[q^3,q,n+2k] qPochhammer[q^2,q^2,n]),
      {k, -n-1, n}, n, 1]

I (18)

qZeil[(1-q^(2k+1)) q^(k(2k+1)) (-1)^k qPochhammer[q,q,n] qPochhammer[-q^2,q,n] *
      qPochhammer[q^3,q^2,n] /
     ((1-q) qPochhammer[q,q,n-2k] qPochhammer[q^3,q,n+2k] qPochhammer[-q^2,q^2,n]),
      {k, -n-1, n}, n, 1]

J (2)

qZeil[(-1)^k q^(k(9k-3)/2) (1+q^(3k)-q^(6k+1)(1-q^(n-3k))/
      (1-q^(n+3k+1))-q^(-3k+1)(1-q^(n+3k))/(1-q^(n-3k+1))) *
      qPochhammer[q,q,2n] qPochhammer[q,q,n-1] /
     (qPochhammer[q,q,n-3k] qPochhammer[q,q,n+3k] qPochhammer[q^3,q^3,n-1]),
      {k, -n, n}, n, 1]

J (3)

qZeil[(-1)^k q^(k(9k-3)/2) (1+q^(3k)-q^(3k)(1-q^(n-3k))/
      (1-q^(n+3k+1))-(1-q^(n+3k))/(1-q^(n-3k+1))) *
      qPochhammer[q,q,2n] qPochhammer[q,q,n-1] /
     (qPochhammer[q,q,n-3k] qPochhammer[q,q,n+3k] q^n qPochhammer[q^3,q^3,n-1]),
      {k, -n, n}, n, 1]

J (4) with q->q^2

qZeil[(-1)^k q^(9k^2) (1-(q^(12k+4)-q^(6k+1))(1-q^(2n-6k))/
      (1-q^(2n+6k+4))+(1-q^(2n+6k+2))/(1-q^(2n-6k+2))) *
      qPochhammer[q^4,q^2,2n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^2,q^2,n-3k] qPochhammer[q^4,q^2,n+3k] qPochhammer[q^3,q^6,n]),
      {k, -n, n}, n, 1]

J (5) with q->q^2

qZeil[(-1)^k q^(9k^2) (q^(6k)-(q^(6k)-q^(12k+3))(1-q^(2n-6k))/
      (1-q^(2n+6k+4))+q^(-6k)(1-q^(2n+6k+2))/(1-q^(2n-6k+2))) *
      qPochhammer[q^4,q^2,2n] qPochhammer[q,q^2,n] /
     (qPochhammer[q^2,q^2,n-3k] qPochhammer[q^4,q^2,n+3k] qPochhammer[q^3,q^6,n] q^(2n)),
      {k, -n, n}, n, 1]

J (6)

qZeil[(1-q^(6k+3)) (-1)^k q^(k(9k+3)/2) qPochhammer[q^3,q,2n] qPochhammer[q,q,n] /
     ((1-q^3) qPochhammer[q,q,n-3k] qPochhammer[q^4,q,n+3k] qPochhammer[q^3,q^3,n]),
      {k, -n-1, n}, n, 1]

L (1) with q->q^4

qZeil[(1+q^(2k)) (-1)^k q^(k(k-1)) qPochhammer[q^4,q^4,n] qPochhammer[-q^2,q^2,2n] /
     (qPochhammer[q^4,q^4,n-k] qPochhammer[q^4,q^4,n+k] (-1)^n q^(2n^2)),
      {k, -n, n}, n, 1]

L (2) with q->q^4

qZeil[(1+q^(2k)) (-1)^k q^(k(2k-1)) qPochhammer[q^4,q^4,2n] /
     (qPochhammer[q^4,q^4,n-k] qPochhammer[q^4,q^4,n+k] qPochhammer[q,q^2,2n]),
      {k, -n, n}, n, 1]

L (3) with q->q^4

qZeil[(1+q^(2k)) (-1)^k q^(k(3k-1)) qPochhammer[q^4,q^4,n] qPochhammer[-q^2,q^2,2n] /
     (qPochhammer[q^4,q^4,n-k] qPochhammer[q^4,q^4,n+k]), {k, -n, n}, n, 1]

L (4) with q->q^4 (corrected I^k ???)

qZeil[(1+q^(4k)) I^k q^(k(k-2)) qPochhammer[-q^2,q^4,n] qPochhammer[q^2,q^2,2n] /
     (qPochhammer[q^4,q^4,n-k] qPochhammer[q^4,q^4,n+k] q^(2n(n-1))),
      {k, -n, n}, n, 1]

L (5) with q->q^2 (corrected)

qZeil[(1+q^(2k)) q^(k(k-1)) qPochhammer[-q,q^2,n]^2 qPochhammer[q,q,2n] /
     (qPochhammer[q^2,q^2,n-k] qPochhammer[q^2,q^2,n+k] qPochhammer[-1,q,2n]),
      {k, -n, n}, n, 1]

L (6) with q->q^2

qZeil[(1+q^(4k)) (-1)^k q^(2k(3k-1)) qPochhammer[-q,q^2,n] qPochhammer[q,q,2n] /
     (qPochhammer[q^2,q^2,n-2k] qPochhammer[q^2,q^2,n+2k]),
      {k, -n, n}, n, 1]

M (1) with q->q^4

qZeil[(-1)^k q^(2k^2) (q^(-k)+q^k) qPochhammer[q^4,q^4,2n] /
     (qPochhammer[q^4,q^4,n-k] qPochhammer[q^4,q^4,n+k] qPochhammer[q,q^2,2n]),
      {k, -n, n}, n, 1]

M (2) with q->q^4

qZeil[(1+q^(2k+1)) q^(k(2k+1)) qPochhammer[q^4,q^4,2n+1] /
     ((1-q^4) qPochhammer[q^4,q^4,n-k] qPochhammer[q^8,q^4,n+k] qPochhammer[-q,q^2,2n+1]),
      {k, -n-1, n}, n, 1]

M (3) with q->q^4 (corrected)

qZeil[(-1)^k q^(2k^2) (q^k-q^(3k+1)) qPochhammer[q^4,q^4,2n+1] /
     ((1-q^4) qPochhammer[q^4,q^4,n-k] qPochhammer[q^8,q^4,n+k] qPochhammer[q,q^2,2n+1]),
      {k, -n-1, n}, n, 1]

below M (3)

qZeil[(1-q^(8k+3)) q^(2k(4k+1)) qPochhammer[q^3,q,2n] /
     (qPochhammer[q^3,q,n+4k+1] qPochhammer[q,q,n-4k] qPochhammer[-q^2,q^2,n]),
      {k, -n-1, n}, n, 1]

Slater [1966]

(3.3.2.1)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[c,q,k] *
      qPochhammer[a d q^n/(b c),q,k] qPochhammer[q^(-n),q,k] qPochhammer[a q/d,q,k] q^k *
      qPochhammer[a q/b,q,n] qPochhammer[a q/c,q,n] qPochhammer[d,q,n] qPochhammer[d/(b c),q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[a q/c,q,k] qPochhammer[d,q,k] *
      qPochhammer[b c q^(1-n)/d,q,k] qPochhammer[a q^(n+1),q,k] qPochhammer[q,q,k] *
      qPochhammer[a q,q,n] qPochhammer[d/c,q,n] qPochhammer[d/b,q,n] qPochhammer[a q/(b c),q,n]),
      {k, 0, n}, n, 1]

(3.4.1.7)

qZeil[qPochhammer[a,q,k] qPochhammer[a q^2,q^2,k] qPochhammer[b,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[a q/b,q,n] qPochhammer[1/(b^2 q),q,n] /
     (qPochhammer[a,q^2,k] qPochhammer[a q/b,q,k] qPochhammer[b^2 q^(2-n),q,k] qPochhammer[q,q,k] *
      qPochhammer[a/b^2,q,n-1] qPochhammer[1/(b q),q,n] (1-a q^(2n-1)/b^2)),
      {k, 0, n}, n, 1]

(3.5.1.5) with a:=q^(2n) (corrected)

qZeil[(-1)^k qPochhammer[q^(2n),q,k] q^(k(3k+1)/2+2n k) / qPochhammer[q,q,k],
      {k, 0, Infinity}, n, 2]

qZeil[qPochhammer[q^(2n+1),q,Infinity] qPochhammer[-q,q,k] q^(k^2+2n k) /
     (qPochhammer[q,q,k] qPochhammer[q^(2n+2),q^2,k]), {k, 0, Infinity}, n, 2]

Srivastava [1983]

(2.4)

qZeil[(-1)^k qBinomial[n,k,q] q^(k(k-1)/2), {k, 0, n}, n, 0]

(2.5)

qZeil[(-1)^k qBinomial[n,k,q] qPochhammer[r/q^k,q,n] q^(k(k-1)/2) /
      qPochhammer[r/q^k,q,1], {k, 0, n}, n, 0]

(A.1)

qZeil[qBinomial[n,k,q] qPochhammer[1/s,q,k] qPochhammer[s,q,n-k] s^k,
      {k, 0, n}, n, 0]

Verma [1977]

between (6) and (7) (corrected)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[q^b,q,k] z^k / (qPochhammer[q^e,q,k] qPochhammer[q,q,k]),
      {k, 0, n}, n, 2, {b, e}]

qZeil[(-1)^n qPochhammer[q^b,q,n] z^n q^(-n(n+1)/2) qPochhammer[q^(-n),q,k] *
      qPochhammer[q^(1-e-n),q,k] (q^(1+e-b+n)/z)^k /
     (qPochhammer[q^e,q,n] qPochhammer[q,q,k] qPochhammer[q^(1-b-n),q,k]),
      {k, 0, n}, n, 2, {b, e}]

(7)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[q^a,q,k] q^(2k) qPochhammer[q^b,q,n] /
     (qPochhammer[q^b,q,k] qPochhammer[q,q,k] qPochhammer[q^(b-a),q,n-1] q^(a(n-1)) *
      (1+q^(n+a)-q^(b-1+n)-q^n)), {k, 0, n}, n, 1, {a, b}]

between (7) and (8) (corrected)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[q^a,q,k] q^(3k) qPochhammer[q^b,q,n] /
     (qPochhammer[q^b,q,k] qPochhammer[q,q,k] qPochhammer[q^(b-a),q,n-2] q^(n(a+2)) *
     (qPochhammer[q^(b-a+n-2),q,2] +
      q^(-1-a-n)(1+q)(1-q^(b-a+n-2))(1-q^n)(1-q^(b+n-1)) +
      q^(-2n-2a) qPochhammer[q^(n-1),q,2] qPochhammer[q^(b+n-2),q,2])),
      {k, 0, n}, n, 1, {a, b}]

(8) (corrected)

qZeil[qPochhammer[q^a,q,k] qPochhammer[q^b,q,k] qPochhammer[q^(-n),q,k] q^k *
      qPochhammer[q^(1+a+b-n-e),q,n] qPochhammer[q^(1+e),q,n] (1-q^(e-b)) /
     (qPochhammer[q^(1+a+b-n-e),q,k] qPochhammer[q^(1+e),q,k] qPochhammer[q,q,k] *
      qPochhammer[q^(1+b-n-e),q,n] q^(n a) qPochhammer[q^(1+e-a),q,n-1] *
      (q^e (1+q^(-e)-q^(-a)-q^n) - q^(e-b)(1-q^(e-a+n)))),
      {k, 0, n}, n, 1, {a, b, e}]

(ii, p.373) (corrected)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b q^(n+1),q,k] qPochhammer[b,q,n+1] /
     (qPochhammer[b,q,k] qPochhammer[q,q,k] qPochhammer[q,q,n] (-1)^(n+1) q^(n(n+1)/2) *
      (b^(n+1) - q^(-n(n+1)))), {k, 0, n}, n, 2]

CheckRec[%, 1]

(iii, p.373) (corrected)

qZeil[qPochhammer[q^(-n),q,k] qPochhammer[b q^(n+2),q,k] qPochhammer[b,q,n+2] /
     (qPochhammer[b,q,k] qPochhammer[q,q,k] qPochhammer[q^2,q,n-1] (-b)^(n+1) *
      q^(n(n+1)/2) (1-q^(2+n)-b q^(n+1)(1-q^(n+1)) +
      (q-1) b^(-n-1) q^(-n(n+1)))),
      {k, 0, n}, n, 2]

CheckRec[%, 1]

(i, p.375)

qZeil[qPochhammer[q^a,q,k] q^k / qPochhammer[q^b,q,k],
      {k, 0, m}, n, 0, {a, b}] /. m->Infinity

Wilf, Zeilberger [1992]

q-Vandermonde-Chu (Sec. 1.6)

qZeil[q^(k^2) qBinomial[n,k,q] qBinomial[b,k,q] / qBinomial[n+b,n,q],
      {k, 0, n}, n, 1, {b}]

q-Saalschuetz (Sec. 1.6)

qZeil[q^((k-n)(k+n)) qPochhammer[q,q,a+b+c-k] /
     (qPochhammer[q,q,a-k] qPochhammer[q,q,b-k] qPochhammer[q,q,c-k] qPochhammer[q,q,k-n] *
      qPochhammer[q,q,k+n] qBinomial[a+b,a+n,q] qBinomial[a+c,c+n,q] *
      qBinomial[b+c,b+n,q]),
      {k, -Infinity, Infinity}, n, 1, {a, b, c}]

q-Dixon (Sec. 1.6)

qZeil[(-1)^k qBinomial[n+b,n+k,q] qBinomial[n+c,c+k,q] *
      qBinomial[b+c,b+k,q] q^(1/2 k(3k-1)) qPochhammer[q,q,n] qPochhammer[q,q,b] *
      qPochhammer[q,q,c] / qPochhammer[q,q,n+b+c],
      {k, -Infinity, Infinity}, n, 1, {b, c}, MagicFactor->-k]

Unknown

Gauss

qZeil[(-1)^k q^((k-n)(k-n-1)) qBinomial[2n+1,k,q] / q^((n+1)n),
      {k, 0, 2n}, n, 1]

CheckRec[%, 1]

q-Analogue of Sum[k k!, {k,0,n}] = (n+1)! - 1

qZeil[qBrackets[k,q] qFactorial[k,q], {k, 0, n}, n, 0]