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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.
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.
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.
There are also verbose versions of the commands which are called by adding one of the following prefixes:
SummaryJust a summary at the end is shown
VerboseSome information in the intermidiate steps
VeryVerboseMore information
ExtraEven more information including information on the linear system in Zeilberger’s algorithm
For example:
GosperVerbose, parGosperVeryVerbose, ZeilbergerExtra,
AntiDifferenceSummary.
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