On 2015-09-07 01:34, Joerg Arndt wrote:
* Bill Gosper <billgosper@gmail.com> [Sep 07. 2015 09:28]:
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),
In George E.\ Andrews, Richard Askey, Ranjan Roy: {Special functions}, Cambridge University Press, (1999). On page 176, "Exercises", relation (1.c) is F([a,b],[2b],x) = (1-x)^{-a/2} *F([a, 2b-a],[b+1/2], - (1-sqrt(1-x))^2 / (4*sqrt(1-x)) ) Swap a and b: F([b,a],[2a],x) = (1-x)^{-b/2} *F([b, 2a-b],[a+1/2], - (1-sqrt(1-x))^2 / (4*sqrt(1-x)) ) Set b = 1/2: F([1/2,a],[2a],x) = (1-x)^{-1/4} *F([1/2, 2a-1/2],[a+1/2], - (1-sqrt(1-x))^2 / (4*sqrt(1-x)) ) This is your transformation.
Best regards, jj
WOW, problem solved! I wonder if you can get this by combining quadratic and linear transformations in http://dlmf.nist.gov/15.8 . Thanks! --rwg (I should check why my methodology that introduced a (matching series coefficients) failed to introduce b.)
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