Fe b 20 08 Searching for Strange Hypergeometric Identities By Sheer Brute

, where one can also find sample input and output. ∞ k=0 (a) k (b) k k!(c) k x k , (where (z) k := z(z + 1)(z + 2) · · · (z + k − 1)), that nowadays is more commonly denoted by 2 F 1 a, b c ; x , has a long and distinguished history, going back to Lehonard Euler and Carl Friedrich Gauss. It was also one of Ramanujan's favorites. Under the guise of binomial coefficient sums it goes even further back, to Chu, in his 1303 combinatorics treatise, that summarized a body of knowledge that probably goes yet further back. The hypergeometric function, and its generalized counterparts, enjoy several exact evaluations, for some choices of the parameters, in terms of the Gamma function. The classical identities of Chu-and others, can be looked up in the classic classic text of Bailey[B], and the modern classic text of Andrews, Askey, and Roy [AAR]. For example, when x = 1, Gauss found the 3-parameter exact evaluation: F (a , b , c , 1) = Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b). (Gauss) (When a is a negative integer, a = −n, then this goes back to Chu's 1303 identity, rediscovered by Vandermonde). Next comes Kummer's two-parameter exact evaluation at x = −1 F (a , b , 1 + a − b , −1) = Γ(1 + a − b)Γ(1 + a/2) Γ(1 + a)Γ(1 + a/2 − b) , (Kummer)