The Summation of Rational Functions by an Extended Gosper Algorithm

The rational functions form the most elementary class of functions for which the problem of summation or antidifferencing is not straightforward. Gosperʹs algorithm finds the antidifference of a rational function only if that antidifference is itself a rational function. We present an algorithm based upon Gosperʹs method which finds the rational part of the antidifference and the purely transcendental summand, i.e., a summand whose antidifference can be expressed entirely as a sum of digamma functions and derivatives of digamma functions. This algorithm is analogous to Horowitzʹs improvement of the Hermite-Ostrogradski method for finding the antiderivative of a rational function. An earlier algorithm of Moenck is the analogue of the Hermite-Ostrogradski method itself. Our algorithm requires less work than Moenckʹs.