This notebook contains the commands from the PhD thesis of Axel Riese
http://www.risc.jku.at/publications/download/risc_118/diss.pdf
in a syntax that corresponds to the current version of the package(s).

<<RISC`qZeil`

<<RISC`pqTelescope`

<<RISC`qSimplify`

Section 1.3.1 Bibasic Summation Formulas
In[2]

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}]

In[3]

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

Section 1.3.2 Bibasic Matrix Inverses
In[4]

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

In[5-7]

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

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

pqTelescope[(1-b/a) (1-a q^k/p^k) (1-b p^k q^k) (-1)^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}]

Section 2.2
In[2-5]

(*tell qZeil to compute the companion identity, too *)
Companion=True;
qZeil[qBinomial[n,k,q] qBinomial[b,k,q] q^(k^2)/qBinomial[b+n,n,q],{k,0,n},n,1,{b}]

(*the companion identity is assigned to CompId*)
CompId

Section 2.3
In[5-8]

(*tell qZeil to compute the dual identity,too*)
Dual=True;
qZeil[qBinomial[n,k,q] qBinomial[b,k,q] q^(k^2)/qBinomial[b+n,n,q],{k,0,n},n,1,{b},Shadow->1]

(*the dual pair is assigned to DualPair*)
DualPair

(*compute the dual identity from the dual pair*)
DualId[{k,0,n},n]

Example 3.2
In[2]

BaileyForw[{(-1)^n q^(n(n-1)/2),KroneckerDelta[n,0]},n]

BaileyForw[%,n]

Theorem 3.6
In[4-6]

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

(*check the m=n case using that the input summand of the last computation has been assigned to FF*)qSimplify[FF/. {k->m,n->m}]

Example 3.3
In[6]

BaileyInv[KroneckerDelta[n,0],n]

In[7]

BaileyInv[1/qPochhammer[q,q,n],n]

In[8]

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

Theorem 3.7
In[9-10]

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

(*check the m=n case;FF is the input summand of the last computation*)qSimplify[FF/. {k->m,n->m}]

Example 3.4
In[11-12]

BaileyBack[{(-1)^n q^(n(n-1)/2),KroneckerDelta[n,0]},n]

BaileyBack[%,n]

Example 3.5
In[13]

BaileyInvBinom[qPochhammer[x,q,n] qPochhammer[q/x,q,n],n,0]

Example 3.7
In[14]

BaileyBackBinom[{(-1)^n (1-q^(2n+1)) q^((c+1)n^2+c n),B},n,1]

Example 3.7
In[15]

BaileyForwBinom[{KroneckerDelta[n,0],qBinomial[2n,n,q]},n,0]

Example 3.8
In[16]

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

Example 3.9
In[17]

BaileyBackBinom[{(-1)^n (1-q^(4n+2)) q^(n(3n+1)),(1-q^(2n+2))*qPochhammer[-q,q,n]},n,2,Step->2]

Example 3.10
In[18]

BaileyDual[{(-1)^n (1+q^(2n)) q^(n(3n-1)),qPochhammer[-q,q,n]/qPochhammer[q,q,2n]},n,Step->2]

Slater's Pairs
In[19]

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

In[20]

MapleCall="maple -q <MapleIn"

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

B.5 The Optional Argument intconst
In[2]

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

B.7.2 Option OnlySummand
In[3-4]

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

qTelescope[%/. {Sigma[0]->1,Sigma[1]->-1},{k,0,n}]

B.7.3 Option MagicFactor
In[5 - 10]

fnk=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]);

qZeil[fnk,{k,0,2n},n,1]

qZeil[fnk,{k,0,2n},n,2]

When trying to compute a recurrence of order 3, it turns out that the underlying system of equations cannot be solved within 90 minutes.

(*
TimeConstrained[qZeil[fnk,{k,0,2n},n,3],90*60,$Failed]
*)

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

qZeil[fnk,{k,0,2n},n,1,MagicFactor->2n-k]

B.7.5 Option FindAlphaBeta
In[11]

h=qZeil[qBinomial[n,k,q]^2/qBinomial[2n,n,q],{k,0,n},n,1,FindAlphaBeta->True]

B.7.6 Option PolyMult
In[12]

qTelescope[(-1)^k q^Binomial[k,2]/(qPochhammer[q,q,n-k] qPochhammer[q,q,n+k]),{k,-n,n},PolyMult->1]

In[14-16]

MapleCall="maple -q <MapleIn";

fnk=qPochhammer[q^(-n),q,k] qPochhammer[c,q,k] qPochhammer[d,q,k] (q^n/(c d))^k/(qPochhammer[q^(1+n),q,k] qPochhammer[q/c,q,k] qPochhammer[q/d,q,k]);

qZeil[fnk,{k,-Infinity,Infinity},n,1]

qZeil[fnk,{k,-Infinity,Infinity},n,1,PolyMult->{2,0}]

qZeil[fnk,{k,-Infinity,Infinity},n,1,PolyMult->{2,1}]

qZeil[fnk,{k,-Infinity,Infinity},n,1,PolyMult->{2,2}]

qZeil[fnk,{k,-Infinity,Infinity},n,1,PolyMult->{2,3}]