A Macsyma implementation of Zeilberger's fast algorithm.

We present the first implementation within the Macsyma computer algebra system of Zeilberger’s fast algorithm for the definite summation problem for a very large class of sequences; i.e. given a hypergeometric sequence F (n, k), we want to represent f(n) = ∑n k=0 F (n, k) in a “simpler” form. We do this by finding a linear recurrence for the summand F (n, k), from which we can obtain a homogeneous k−free recurrence for f(n). The solution of this recurrence is left as a post-processing, and it will give the “simpler” form we were looking for. Zeilberger’s fast algorithm exploits a specialized version of Gosper’s algorithm for the indefinite summation problem; i.e. given a hypergeometric sequence t(k), the problem of finding another sequence T (k) such that t(k) = ∆kT (k) = T (k + 1)− T (k). The implementation of this algorithm has also been carried out in Macsyma, and its details are also briefly described in this paper.