OOPS!! Pofessor A. B. Olde Daalhuis, DLMF editor for Hypergeometric Functions (Chapters 13, 15, 16), kindly informs me that this transformation follows from switching the sides of http://dlmf.nist.gov/15.8.E22 and recanonicalizing the argument and parameters! I typed in all of 15.8, but I should have been more vigilant. Adding to my confusion, FullSimplify demurs due to the apparent failure of the reversed identity on the negative real axis. We may need one of Marichev's patented Sqrt[x]*Sqrt[1/x] fudge factors. --rwg On 2015-09-07 02:04, rwg wrote:
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