aghitza / hipergeo

Hypergeometric package for Maxima featuring Sister Celine, Gosper, Zeilberger, WZ, and WZ Certificate.

Suppose your career depends on solving sums like

or more intimidating ones as

and your magic tricks toolbox fail to handle it. What is the last resort?

Synopsis

I developed a Maxima package to either solve hypergeometric sums or prove they cannot be solved.

Description

hipergeo is a hypergeometric package for the computer algebra system Maxima featuring:

• Sister Celine implementation;
• friendly link to Gosper implementation from `zeilberger' package;
• Builds full recurrence from Zeilberger's algorithm;
• Wilf-Zeilberger proof machinery along with WZ certificate implementation.

For Portuguese readers, you can refer to base text for usage, algorithms, theory, etc. Here you will have a pragmatic introduction to hypergeometric summation, 100% Maxima-oriented.

As for English readers, you can refer to the seminal A=B book by Petkovsek, Wilf and Zeilberger for more details on the algorithms and a deeper insight into the theory. (This book uses Maple and Mathematica computer algebra systems rather than open source Maxima.)

Using the library

In Maxima, type

> load('hipergeo.mac');

in order to load the package and browse the examples in the same directory.

First example can merely be solved by first finding the anti-difference of the summand,

> summand(k) := k*k!;
> antidiff : gosper(summand, k);
k!

and, by applying the fundamental theorem of calculus, that sum becomes n! - 1.

Any help is appreciated, should you find any error please contact or send a pull request and I will be delighted to cite your amendment.