Summation of binomial coefficients using hypergeometric functions

Abstract

An algorithm which finds the definite sum of many series involving binomial coefficients is presented. The method examines the ratio of two consecutive terms of the series in an attempt to express the sum as an ordinary hypergeometric function. A closed form for the infinite sum may be found by comparing the resulting function with known summation theorems. It may also be possible to identify ranges of the summation index for which summing to a finite upper limit is the same as summing to infinity.