87. Binomial Coefficient Identities and Hypergeometric Series

In recent months I have come across many instances in which someone has found what they believe is a new result, in which they evaluate in closed form a sum involving binomial coefficients or factorials. In each case they have managed to do that either by using the recent powerful method of Wilf and Zeilberger (the W–Z method) [6], or by comparing coefficients in some ad hoc algebraic identity. The aim of this note is to describe, using a few examples, a purely algorithmic method for re–casting the sum as a (multiple of a) hypergeometric series in standard notation, so that one can then simply look up standard tables of hypergeometric series to see if the series under investigation is “summable” via known results. I do not claim any originality in this idea. I got it from Richard Askey, who claims [2] that “at least 90% if not 95% of the formulas in Table 3 of Henry Gould’s Tables [3] yield to this approach”. Indeed, to quote Askey further, “For years [before working with George E. Andrews in 1973] I had been trying to point out that the rather confused world of binomial coefficient summations is best understood in the language of hypergeometric series identities. Time and again I would find first–rate mathematicians who had never heard of this insight and who would waste considerable time proving some apparently new binomial coefficient summation which almost always turned out to be a special case of one of a handful of classical hypergeometric identities.” The identities I will use to illustrate the method are the following. The first came to me in a paper I was asked to referee, but is to be found in Wang and Guo [8] (1989). The second was found in 2001 by an Honours student at UNSW, T. T. To [7], in his fourth–year essay, and the third appeared in a very recent article by Victor Moll [5] in the Notices of the American Mathematical Society. They are