(maxima.info)Introduction to zeilberger

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89.1 Introduction to zeilberger
===============================

'zeilberger' is a 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.

89.1.1 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.

89.1.2 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.

89.1.3 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'.