I was bugging the DLMF hypergeometric people about having only eight special values (Ira Gessel points out that one of them is redundant), and claimed there should be at least a hundred. Just now I notice http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/ claiming a completely incredible 111271 formulas. The obvious challenge: Find one they missed. Or vastly preferable: an independent contiguous pair. Missing from http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/05... Hypergeometric2F1[1/2, 3 a, 1/2 + 2 a, -1/3] == ((27/4)^a Gamma[1 + a] Gamma[1/2 + 2 a])/(Sqrt[\[Pi]] Gamma[1 + 3 a]) and Hypergeometric2F1[1/2, 1 + 3 a, 3/2 + 2 a, -1/3] == 4^-a Sqrt[\[Pi]/3] Gamma[3/2 + 2 a]/Gamma[1 + a]^2 . --rwg