Duh, the original argument was < -1, so the z/(z-1) transformation gives a single, convergent series: 10 3 1/12 1/12 13 1 5 2 3 5 sqrt(6) 17 23 hyper_2f1(--, --, -, --------) = --------------------- 24 24 6 2 2 1/6 17 23 3125 --rwg On Thu, Nov 3, 2011 at 12:23 AM, Bill Gosper <billgosper@gmail.com> wrote:
Rich pointed me to ( http://arxiv.org/abs/1110.6284 , 37kb) whose abstract ends: "An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities."
Note the argument > 9.4. Empirically, the two 2F1s produced by the 1/z transformation satisfy
Hypergeometric2F1[1/24, 5/24, 3/ 4, -(14641/138240)] == ((45 + 26 Sqrt[3])^(1/3) Gamma[7/12] Gamma[3/4])/ (2^(1/6) 3^(1/24) 5^(19/24) Sqrt[\[Pi]] Gamma[5/6])
and
Hypergeometric2F1[7/24, 11/24, 5/4, -(14641/138240)] == (2 3^(17/24) (2 (-45 + 26 Sqrt[3]))^(1/3) Sqrt[\[Pi]] Gamma[1/12])/ (11 5^(13/24) Gamma[3/4] Gamma[5/6])
which together imply the "denesting"
-(3 (-45 + 26 Sqrt[3]))^(1/3) + (3 (45 + 26 Sqrt[3]))^(1/3) == 6
--rwg