Oops, of course I meant parameters mod 1. Can we say for sure whether or not 2F1[1/3,2/3,1,z] or 2F1[1/6,2/3,1,z] or 2F1[1/6,5/6,1] is algebraic K[algebraic]? (Mod some factors of π.) Glad to see Mathematica is learning about contiguity! --rwg To do this, Mathematica must know at least two contiguous identities. If there's a symbolic parameter a, say, you can manufacture a contiguous identity just by incrementing a, but the contiguity recurrences may only provide a sublattice of valuations due to division by 0, so you need a pair of "independent" valuations. Below are two such pairs, of which Mathematica and Functions.wolfram.com seem completely unaware. I can supply a number of others which can be hard to find with or without a library. Out[1026]= {Hypergeometric2F1[a, 1/2 + a, 3/2 - 4 a, 1/5] == ( 2^(3/2 - 10 a) 5^(-(3/2) + 6 a) Sqrt[(5 + Sqrt[5]) \[Pi]] Gamma[3/2 - 4 a])/(Gamma[4/5 - 2 a] Gamma[6/5 - 2 a]), Hypergeometric2F1[a, 1/2 + a, 5/2 - 4 a, 1/5] == ( 2^(7/2 - 10 a) 5^(-(5/2) + 6 a) Sqrt[(5 - Sqrt[5]) \[Pi]] Gamma[5/2 - 4 a])/(Gamma[7/5 - 2 a] Gamma[8/5 - 2 a]), Hypergeometric2F1[5 a, 1/2 + 5 a, 9/10 + 4 a, 1/5] == ( 2^(3/10 - 2 a) Sqrt[\[Pi] - \[Pi]/Sqrt[5]] Gamma[9/10 + 4 a])/( Gamma[3/5 + 2 a] Gamma[4/5 + 2 a]), Hypergeometric2F1[5 a, 1/2 + 5 a, 7/10 + 4 a, 1/5] == ( 2^(-(1/10) - 2 a) Sqrt[(1 + 1/Sqrt[5]) \[Pi]] Gamma[7/10 + 4 a])/( Gamma[2/5 + 2 a] Gamma[4/5 + 2 a])} Specializations to make an EllipticK: In[1027]:= FunctionExpand[% /. a -> 1/8] Out[1027]= {( 2 (2/(1 + 2/Sqrt[5]))^(1/4) EllipticK[1/2 - 1/(5^(1/4) Sqrt[1 + 2/Sqrt[5]])])/\[Pi] == ( 11 Sqrt[(5 + Sqrt[5]) \[Pi]])/( 10 10^(3/4) Gamma[19/20] Gamma[31/20]), ( 64 5^(3/4) (2 (1 + 2/Sqrt[5]))^(1/4) EllipticE[1/2 - 1/(5^(1/4) Sqrt[1 + 2/Sqrt[5]])])/(21 \[Pi]) - ( 32 (2/(1 + 2/Sqrt[5]))^(1/4) EllipticK[1/2 - 1/(5^(1/4) Sqrt[1 + 2/Sqrt[5]])])/(21 \[Pi]) - ( 32 Sqrt[5] (2/(1 + 2/Sqrt[5]))^(1/4) EllipticK[1/2 - 1/(5^(1/4) Sqrt[1 + 2/Sqrt[5]])])/(21 \[Pi]) - ( 32 5^(3/4) (2 (1 + 2/Sqrt[5]))^(1/4) EllipticK[1/2 - 1/(5^(1/4) Sqrt[1 + 2/Sqrt[5]])])/(21 \[Pi]) == ( 4 2^(1/4) Sqrt[(5 - Sqrt[5]) \[Pi]])/( 5 5^(3/4) Gamma[23/20] Gamma[27/20]), -(8/ 99) (913 Hypergeometric2F1[-(7/8), 5/8, 2/5, 1/5] + 945 Hypergeometric2F1[-(3/8), 1/8, 2/5, 1/5] - 1526 Hypergeometric2F1[1/8, 5/8, 2/5, 1/5]) == -(( 4 2^(1/20) Sqrt[5 (5 - Sqrt[5]) \[Pi]] Gamma[7/5])/( 3 Gamma[-(3/20)] Gamma[21/20])), -(4/ 51) (237 Hypergeometric2F1[-(7/8), 5/8, 1/5, 1/5] + 1445 Hypergeometric2F1[-(3/8), 1/8, 1/5, 1/5] - 1366 Hypergeometric2F1[1/8, 5/8, 1/5, 1/5]) == -(( 2 2^(13/20) Sqrt[5 (5 + Sqrt[5]) \[Pi]] Gamma[6/5])/( 7 Gamma[-(7/20)] Gamma[21/20]))} In[1028]:= FunctionExpand[%% /. a -> 1/40] Out[1028]= {-(16/ 5) (5 Hypergeometric2F1[-(39/40), 21/40, 2/5, 1/5] - 4 Hypergeometric2F1[1/40, 21/40, 2/5, 1/5]) == ( 2 2^(1/4) Sqrt[(5 + Sqrt[5]) \[Pi]] Gamma[7/5])/( 5 5^(7/20) Gamma[3/4] Gamma[23/20]), 256/55 (3 Hypergeometric2F1[-(39/40), 21/40, 2/5, 1/5] - 2 Hypergeometric2F1[1/40, 21/40, 2/5, 1/5]) == ( 8 2^(1/4) Sqrt[(5 - Sqrt[5]) \[Pi]] Gamma[12/5])/( 25 5^(7/20) Gamma[27/20] Gamma[31/20]), ( 2 (2/(1 + 2/Sqrt[5]))^(1/4) EllipticK[1/2 - 1/(5^(1/4) Sqrt[1 + 2/Sqrt[5]])])/\[Pi] == -(( 4 2^(1/4) Sqrt[5 (5 - Sqrt[5]) \[Pi]])/( 7 Gamma[-(7/20)] Gamma[17/20])), Hypergeometric2F1[1/8, 5/8, 4/5, 1/5] == -(( 2 2^(17/20) Sqrt[5 (5 + Sqrt[5]) \[Pi]] Gamma[4/5])/( 11 Gamma[-(11/20)] Gamma[17/20]))} On Sat, Aug 8, 2015 at 8:37 PM, Oleksandr Pavlyk <pavlyk@wolfram.com> wrote:
On 8/8/2015 10:19 PM, Bill Gosper wrote:
2F1[rational,rational,1 or 1/2,x] ==algebraic1[x] EllipticK[algebraic2[x]] identities? Or at least any more than a,b,c =
Did you try
FunctionExpand[Hypergeometric2F1[1/4 - 2, 1/4 + 1, 1 + 3, x]]
FunctionExpand[Hypergeometric2F1[1/4 - 2, 3/4 + 1, 1/2 + 3, x]]
FunctionExpand[Hypergeometric2F1[1/8 + 4, 5/8 - 2, 1 + 1, x]]
etc.
--Sasha
1/2,1/2,1 1/4,1/2,1 1/4,3/4,1 1/4/1/4,1 1/4,1/4,1/2 1/4,3/4,1/2 (sort of) 1/8,5/8,1 1/12,7/12,1 ? This last one features seriously unwieldy cubics. Tnx, --rwg