Computer Programs From the Book "A=B", and Related Programs
recent update: September 13, 2000] is a package of Maple programs for
recurrences satisfied by hypergeometric sums, including Zeilberger's
telescoping" algorithm, Sister Celine's algorithm, etc. The most recent
updates have been for the purpose of adapting the programs to later
releases of Maple.
is similar to EKHAD, but deals with q-identities.
- While you're in the neighborhood, be sure to visit Doron
Zeilberger's home page where you will find a great deal of related
is the Harvard minor thesis of Brad Mann, which is an elegant
exposition of the ideas of "A=B" from an interesting vewpoint.
- Hyper is a
program that will find hypergeometric solutions of linear recurrences
polynomial coefficients, if they exist, or will prove that they do not
exist, if they don't.
of Hyper - it finds all q-hypergeometric solutions of q-difference
with rational coefficients.
is a Mathematica
package that will execute Gosper's algorithm for indefinite
summation, as well as the WZ algorithm, the GosperFunction routine, and
FactorialSimplify. As a bonus, this package contains "MixedZeil", which
is a beta version of a multibasic creative telescoping algorithm. Your
comments are invited. [July 11, 1996]
- WZ is a
that finds WZ proof certificates for hypergeometric sums (requires
- The implementation of Zeilberger's algorithm in Mathematica,
package zb.m of Peter Paule and Markus Schorn, is available
However this page is password protected. To get the password, send an e-mail request.
- FindRecur is a Mathematica program that
carries out Sister Celine's algorithm. The one that is given on
page 61 of A=B does not function properly with recent versions of
Mathematica. This updated version of FindRecur was contributed by
Professor Erich Neuwirth of the University of Vienna, to whom our
thanks are due.
(1) For best results, use the Mathematica SetDirectory
to set the directory to the one that holds the Zb routines on your
e.g., SetDirectory["/myprograms/zb"]. This command can be
at the beginning of the package zb.m, and then the job will
done automatically each time you load the programs.
Here is a link to the home page of Christian Krattenthaler, from
you will be able to obtain the programs "HYP"
(2) If you are using Mathematica version 3.0 or later and
have already downloaded these packages, then you will have difficulty
you do the following: In your favorite text editor, edit the program zeilutil.m.
Globally search and replace every occurrence of the string "Split"
by the string "Ssplit". Then edit the program algutil.m
and do the same replacement. (Alternatively, just download the updated
versions of all of these programs from the Web site at RISC-LINZ).
New and very powerful software
for multisums has been developed by Kurt Wegschaider, of RISC-LINZ.
His refinement of Sister Celine's method, in the multivariate case,
surprisingly quickly on some very difficult problems. Be sure to read
diploma thesis, available from the same page, which describes his ideas
and the implementation, with many examples.
On Bruno Gauthier's web site you'll find his Maple package HYPERG,
which deals with hypergeometric series, solving recurrences, Gosper's
Zeilberger's algorithms, etc.
A REDUCE package, ZEILBERG, written by
(firstname.lastname@example.org) is part of the new REDUCE
3.6, and the Maple package sumtools will be part of the next
release. Furthermore, Maple's sum command will include
invoking this package by an implementation of Wolfram Koepf. Wolfram
has published a book Hypergeometric Summation.