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.

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Abstract. Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the confluent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the abovementioned extended functions and investigating various formulas for the further extended functions in a systematic manner. Moreover, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

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Motivated by the extension of classical Gauss's summation theorem for the series . As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.F