I was going to (respectfully) ask who the heck cares about 2F1[1/2,5/12,1,x] but I see my notes imply Hypergeometric2F1[1/12, 5/12, 1, (27 (-1 + z) z^2)/(-4 + 3 z)^3] == ( Sqrt[2] (8 - 9 z)^(1/6) (-((-4 + 3 z)^3/(8 - 9 z)^2))^(1/12) EllipticK[2 - 2/(1 + Sqrt[z])])/(\[Pi] Sqrt[1 + Sqrt[z]]) which Plot3D says holds everywhere but z>1. Maier seductively suggests Hypergeometric2F1[1/12, 5/12, 1, ( 108 (-1 + z) z)/(1 + 16 (-1 + z) z)^3] == ( 2 ((1 + 16 (-1 + z) z)^3)^(1/12) EllipticK[z])/\[Pi] but this only holds in a tiny (r~.05) off-center ellipse. Thanks! --rwg On 2015-10-07 08:31, Joerg Arndt wrote:
The paper Robert S.\ Maier: {Algebraic hypergeometric transformations of modular origin}, Transactions of the American Mathematical Society, vol.359, no.8, pp.3859-3885, (August-2007). http://www.ams.org/journals/tran/2007-359-08/S0002-9947-07-04128-1/ gives formulas (relations (6.1.*) on page 3875) to transform F([s, s], [1], x) to and from F([1/2,5/12],[1], x) for s in {1/2, 1/3, 1/4} (reproduced on page 614 of the fxtbook). s = 1/2 corresponds to ellipticK, s=1/3 (after Pfaff reflection) to your F([1/3,2/3],[1],x).
More (quality) articles by Maier: http://arxiv.org/find/math/1/au:+Maier_R/0/1/0/all/0/1
For more material I'd suggest to check articles of Garvan, Berndt, Borwein (those with "cubic" in title or abstract) and follow the trail of citations (to and from then). This will be quite some work! Sure there are more authors that should be named, suggestions welcome.
Best regards, jj
* Bill Gosper <billgosper@gmail.com> [Oct 07. 2015 15:26]:
Tabulating 2F1s without symbolic parameters nor argument makes about as much sense as trying to build a bestiary with complete genomes. Unless the table is gigantic, the chance of matching your exact animal is minute. Nevertheless, you might enjoy a few closed forms, simultaneously expressed as 2F1[1/3,2/3,1,<algebraic>], Gammas, and 2F1[1/2,1/2,1,<algebraic>] = K(<algebraic>). --rwg gosper.org/sig2sig3.png (Still no luck simplifying that ridiculous expression for 2F1[1/3,2/3,1,-1].)
You might not believe the attacks it has resisted.