An extension of Zeilberger's fast algorithm to general holonomic functions
In this work, an algorithm is given for the summation of holomonic sequences, therefore generalizing Zeilberger's algorithm which works for hypergeometric sequences. Whereas Zeilberger's approach is a clever application of Gosper's algorithm, in the current paper an algorithm of Abramov for the computation of rational solutions of recurrence equations with polynomial coefficients is used. With this approach, identities like \[ \sum_{k=0}^n\Biggl(\sum_{j=0}^k{n \choose j}\Biggr)^3= n2^{3n-1}+2^{3n}-3n2^{n-2} {2n \choose n} \] can be computer-generated. The technique can be extended to the differential case and to \(q\)-calculus.
scientific article; zbMATH DE number 1453348