Generalizations of Whipple's theorem on the sum of a 3F2

Abstract

By systematically applying ten inequivalent two-part relations between hypergeometric sums 3 F 2 (…|1) to the published database of all such sums, 66 new sums are obtained. Many results extracted from the literature are shown to be special cases of these new sums. In particular, the general problem of finding elements contiguous to Watson's, Dixon's and Whipple's theorem is reduced to a simple algorithm suitable for machine computation. Several errors in the literature are corrected or noted.

An Extension of the Triple Hypergeometric Series by Exton

“…In 1996, Lavoie, Grodin and Rathie [8] have given the generalization of Kummer's theorem on the sum of a 2 F 1 and obtained ten results, in the form of a single result…”

“…The coefficients A i and B i are given in [8]. We start with Dixon's classical result [1, p. 13, Eq.…”

“…The coefficients A i,j and B i,j are given in the tables at the end of the paper [7]. More specifically, 39 distinct formulas are given, including Dixon's result which appears when i = j = 0.…”

“…[12]. In 2007, Kim and Rathie [5] derived twenty five transformation formulas in the form of a single result for X 8 with the help of the generalized Watson's theorem [9]. We obtained 25 results for the series :…”

APPELL'S FUNCTION F1AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X9

“…It should also be remarked here that whenever hypergeometric and generalized hypergeometric functions reduce to express in terms of Gamma functions, the results are very important from the applicative point of view. Therefore, the classical summation theorems such as those of Gauss, Gauss's second, Bailey and Kummer for the series 2 F 1 and Dixon, Watson, Whipple and Saalschütz for the series 3 F 2 and their rather recent extensions and generalizations (see [7], [8], [9], [10] and [11]) play an important role in the theory of hypergeometric and generalized hypergeometric series. For applications of the above-mentioned classical summation theorems, we refer to [2], [5], [6], [10], [11], [12] and [13].…”