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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:
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
.
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