77.1 Introduction to zeilberger

zeilberger is an implementation of Zeilberger’s algorithm for definite hypergeometric summation, and also Gosper’s algorithm for indefinite hypergeometric summation. zeilberger makes use of the "filtering" optimization method developed by Axel Riese. zeilberger was developed by Fabrizio Caruso. load("zeilberger") loads this package.

77.1.1 The indefinite summation problem

zeilberger implements Gosper’s algorithm for indefinite hypergeometric summation. Given a hypergeometric term F_k in k we want to find its hypergeometric anti-difference, that is, a hypergeometric term f_k such that F_k = f_(k+1) - f_k.

77.1.2 The definite summation problem

zeilberger implements Zeilberger’s algorithm for definite hypergeometric summation. Given a proper hypergeometric term (in n and k)

F_(n,k) and a positive integer d we want to find a d-th order linear recurrence with polynomial coefficients (in n) for F_(n,k) and a rational function R in n and k such that

a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),

where Delta_k is the k-forward difference operator, i.e., Delta_k(t_k) := t_(k+1) - t_k.

77.1.3 Verbosity levels

There are also verbose versions of the commands which are called by adding one of the following prefixes:

Summary

Just a summary at the end is shown

Verbose

Some information in the intermidiate steps

VeryVerbose

More information

Extra

Even more information including information on the linear system in Zeilberger’s algorithm

For example:

GosperVerbose, parGosperVeryVerbose, ZeilbergerExtra, AntiDifferenceSummary.