Introduction to zeilberger (Maxima 5.47.0 Manual)

95.1 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.

Categories: Sums and products · Share packages · Package zeilberger ·

The indefinite summation problem
The definite summation problem
Verbosity levels

95.1.1 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

\[F_k = f_{k+1} - f_k. \]

95.1.2 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

\[a_0 \, F_{n,k} + \ldots + a_d \, F_{n+d,k} = \Delta_K \left(R\left(n,k\right) F_{n,k}\right), \]

where \(\Delta_k\) is the \(k\)-forward difference operator, i.e., \(\Delta_k \left(t_k\right) \equiv t_{k+1} - t_k.\)

95.1.3 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 intermediate steps
VeryVerbose: More information
Extra: Even more information including information on the linear system in Zeilberger’s algorithm

For example:
GosperVerbose, parGosperVeryVerbose, ZeilbergerExtra, AntiDifferenceSummary.