I have the weird feeling I once knew this and forgot it, but since I no longer have a decent message massager (or "masausage", as a young friend recently misspoke) we'll have to risk repetition. 1 - sqrt(5) hyper_2f1(a, 5 a + 1, 3 a + 1, -----------) = 2 5 a --- 2 3 2 pi 5 (a - -)! (a - -)! 4 %pi cos(--) 5 5 10 (3 a)! (---------------------- + ----------------------------) pi 5 a (5 a)! 4 %pi cos(--) --- 10 2 3 2 5 a! 5 (a - -)! (a - -)! 5 5 Note G/(5a)! + 1/G/5a!. I think this is almost, but not quite derivable from the formulas in A&S Chapter 15, starting with 15.1.31, and then various linear and quadratic transformations. The obstacle seems to be A&S's negligence to provide contiguous pairs both for 15.1.31 and the quadratic transformations. The latter can be somewhat tediously derived from the three-term recurrences, but 15.1.31 seems to be cursed with some sort of singularity that thwarts population of the contiguity grid starting with F(a) and F(a+-3). (Of course, the good way to do things is with 3x3 matrices that have the contiguous pair built in. I'm belatedly nagging the DLMF people about this.) There are also similar identities for 2F1(phi^-2). Once all this goes into the HYPERSIMP facility, we'll see whether 2F1(a,5a,3a,1/phi) or some contiguous neighbor has a monomial rhs. --rwg