Also, (17+21√5+I√3 (17-7√5))^(1/4)==((-1-I√3+3√5-I√15)√I)/(2√2) --rwg (Good grief, there's a unicode for ༔ ?) On Mon, Nov 27, 2017 at 5:28 PM, Bill Gosper <billgosper@gmail.com> wrote:
Stumbling through old mail I was startled by
Hypergeometric2F1[1/3, 2/3, 1, 1 - GoldenRatio] == 2 Sqrt[GoldenRatio] π/(5^(7/12) Gamma[1/3] Gamma[11/15] Gamma[14/15])
I have no idea where I got it, but a fairly immediate consequence is
EllipticK[1/64 (47 + 17 I Sqrt[3] - 21 Sqrt[5] - 7 I Sqrt[15])] == ((1 + Sqrt[5]) (17 + 17 I Sqrt[3] + 21 Sqrt[5] - 7 I Sqrt[15])^(1/4) π^2)/( 2 Sqrt[2] 3^(3/4) 5^(7/12) Gamma[1/3] Gamma[11/15] Gamma[14/15])
I don't have any η(i√15)s in my identity collection, but I strongly suspect Gamma[1/3] Gamma[11/15] Gamma[14/15] will be Chowla-Selberg's prediction for the transcendental part. The algebraic part was mostly E^(I ArcCot[1/8 (1 + Sqrt[3])^4 (Sqrt[3] + Sqrt[5])^2]) . (Huh? That's algebraic? Obviously!) --rwg Many of these 2F1s come out as algebraic*ElliptickK[algebraic] π^rational. MROB: Let me know if you ever add K to ries. --Bill