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

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

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

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.