The elliptic K transformation that NeilB & I are investigating generalizes (from a=1/2) at least to (*) Hypergeometric2F1[a,1/2, 2 a, z] == Hypergeometric2F1[2 a - 1/2, 1/2, 1/2 + a, -((-1 + Sqrt[1 - z])^2/(4 Sqrt[1 - z]))]/(1 - z)^(1/4), which strongly resembles published 2F1 quadratic transformations (e.g. http://dlmf.nist.gov/15.8), except these generally have two degrees of freedom, e.g. 2F1[a,b,c(a,b),z] = f(z,a,b) 2F1[A(a,b),B(a,b),C(a,b),g(z)], where f and g are algebraic in z. DLMF lists an exception: "When the intersection of two groups in Table 15.8.1 <http://dlmf.nist.gov/15.8#T1> is not empty there exist special quadratic transformations, with only one free parameter, between two hypergeometric functions in the same group. . . . For further examples see Andrews et al. (1999 <http://dlmf.nist.gov/bib/#bib102>, pp. 130–132 and 176–177)." Can someone with access peek at http://www.ams.org/mathscinet-getitem?mr=1688958 (http://ebooks.cambridge.org/ebook.jsf?bid=CBO9781107325937) and tell us if (*) appears there? Or do you know of other such single degree of freedom formulæ? Or best of all, can you find the 2F1[a,b,...] generalization of (*) that would finally put this question to rest? --rwg