SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #13 --------------- OP-SF NET --------------- September 12, 1996

(q)-Zeilberger algorithm

From: Tom Koornwinder

Below follows an update about implementations of the Zeilberger algorithm and its q-analogue, and accompanying papers and books.

1. The most recent versions of Doron Zeilberger's own Maple implementations (see OP-SF Net 1.1, Topic #14) are EKHAD and qEKHAD, obtainable by anonymous ftp from, directory pub/zeilberg/programs or via Doron's home page
An accompanying book, in fact covering much more appeared this summer:
M. Petkovsek, H.S. Wilf & D. Zeilberger, A=B, A.K. Peters, 1996.

2. Tom Koornwinder's Maple implementations zeilb and qzeilb (see OP-SF Net 1.1, Topic #14) dating back from 1992 have just been slightly revised and adapted to Maple V, Release 4. They are obtainable by anonymous ftp from, directory
or via Tom's home page The accompanying paper

T.H. Koornwinder,
On Zeilberger's algorithm and its q-analogue,
J. Comput. Appl. Math. 48 (1993), 91-111

has been adapted accordingly. The slightly revised version, with title "On Zeilberger's algorithm and its q-analogue: a rigorous description" is also available from the ftp site just mentioned.

3. Wolfram Koepf implemented Zeilberger's algorithm and certain extensions in Maple V, Release 4. (See OP-SF Net 1.9, Topic #5 for an earlier implementation by him in 1994.) The present implementation is part of the official distribution of Maple V, Release 4. It can be made operational by the two commands

> with(sumtools):
> readlib(`sum/simpcomb`):

possibly followed, to get help, by

> ?sumtools

The accompanying paper is:

W. Koepf,
Algorithm for m-fold hypergeometric summation,
J. Symb. Comput. 20 (1995), 399-417.

Wolfram Koepf also wrote a book manuscript about these topics:

W. Koepf,
Algorithmic Summation and Special Function Identities with Maple,
To appear.

The new package "code", to be used for the generation of recurrence and differential equations for sums and integrals, written in connection with this book can be obtained from his home page

4. Peter Paule and M. Schorn implemented Zeilberger's algorithm in Mathematica (see OP-SF Net 1.3, Topic #10), while Peter Paule and Axel Riese similarly implemented the q-Zeilberger algorithm. These implementations are available on email request to Peter Paule. The accompanying papers are:

5. Rene Swarttouw used Koepf's Maple implementations of the Zeilberger algorithm for an interactive package on World Wide Web for calculating formulas for orthogonal polynomials belonging to the Askey-scheme. See WWW:
You can also approach this via Rene's home page

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