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.