If you split Ramanujan's 1/pi = 4F3[1/64] (search for Ramanujan in www.tweedledum.com/rwg/idents.htm) into two 3F2s, Mma 6.0 recognizes one of them, producing the identity expand(Hyper_F[P,Q]([1,1,3]/2,[1,1],1/64)=16/(21*%Pi) + (64*Elliptic_Kc(1/2*(1-3*Sqrt(7)/8))^2)/(21*%Pi^2)) 3 1 1 1 hyper_f ([-, -, -], [1, 1], --) = 3, 2 2 2 2 64 2 1 3 sqrt(7) 64 elliptic_kc (- - ---------) 2 16 16 ------------------------------ + ------ 2 21 %pi 21 %pi ~ 1.01336493939236. If we can find one more, we might get the whole contiguous family, since 3F2[z] obeys four term recurrences. Embarrassments: Mma 6.0 can't do FullSimplify[ FunctionExpand[ HypergeometricPFQ[{1, a, b}, {c, 3 - c + b + a},1]]] nor even the special case HypergeometricPFQ[{1,1,1},{5,5}/2,1] = 9 pi - 27, even though the general sum telescopes. My enhanced Macsyma also fails unless you first convert to Sigma notation (makesum). --rwg PS, overall, I'm favorably impressed with Mma 6.0, although I did lose a session when something in the help system cleared all my definitions.