Engel Expansion

File: EngelManual.nb
Author: Burkhard Zimmermann,
last modification: November 17, 2014.

This file supplements the paper "A Mathematica Package for Extended Engel Expansion of q-Series" by G.E.Andrews, A. Knopfmacher, P.Paule and B.Zimmermann, but uses a input syntax appropriate for version 1.12 of the Engel package.

Loading Engel.m

<<RISC`Engel`

Package Engel version 1.12 written by Burkhard Zimmermann
(in cooperation with G.E. Andrews, A. Knopfmacher, and P. Paule)
Revised in 2008, 2009, 2011, 2014 by Ralf Hemmecke.
Contains qNormal.m, written by Axel Riese.
Copyright 2000-2014, Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

Examples

the Rank Generating Function

A = 1+Sum[t^(k-1)*q^k*qPochhammer[-q/t,q,k-1],{k,1,20}];

Engel[A,q,12]

Engel[A,q,12,EngelSeriesOutput->True]

an identity of Euler

A=qPochhammer[-z q,q];

Engel[A,q,10]

Engel[A,q,10,EngelSeriesOutput->True]

an identity of Euler,  special case z=1, ExtraExponent->1

Engel[qPochhammer[-q,q],q,20,ExtraExponent->1]

Engel[qPochhammer[-q,q],q,20,ExtraExponent->1,EngelSeriesOutput->True]

an identity of Euler,  special case z=1, ExtraExponent->2

Engel[qPochhammer[-q,q],q,40,ExtraExponent->2]

an identity of Euler,  special case z=1, ExtraExponent->2 -- companion

Engel[(1-q)*qPochhammer[-q,q],q,40,ExtraExponent->2]

Engel[(1-q)*qPochhammer[-q,q],q,40,ExtraExponent->2,EngelSeriesOutput->True]

An identity of Cauchy

Engel[1/qPochhammer[z*q,q],q,10,ExtraExponent->1]

Engel[1/qPochhammer[z*q,q],q,10,ExtraExponent->1,EngelSeriesOutput->True]

identity (11) on Slater's list.

qx[n_]:=qPochhammer[q^n,q^16]*qPochhammer[q^(16-n),q^16];

Engel[1/(qx[2]*qx[3]*qx[4]*qx[5]),q,40,ExtraExponent->1]

Engel[1/(qx[2]*qx[3]*qx[4]*qx[5]),q,40,ExtraExponent->1,EngelSeriesOutput->True]

identity 12 on Slater's list.

qx[n_]:=qPochhammer[q^n,q^16]*qPochhammer[q^(16-n),q^16];

Engel[(1-q)/(qx[1]*qx[4]*qx[6]*qx[7]),q,40,ExtraExponent->1]

the First Rogers Ramanujan identity.

Engel[1/qPochhammer[q,q^5]/qPochhammer[q^4,q^5],q,20]

the number of partitions of n

Engel[1/qPochhammer[q,q],q,15,ExtraExponent->1]