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http://functions.wolfram.com/07.23.03.a8sg.01
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Hypergeometric2F1[-(21/4), 15/4, 11/2, z] ==
(1/(1342494153 Pi^(3/2) z^(9/2)))
(8 (-2 Sqrt[z] (-185640 + 318240 z + 272493 z^2 + 487305 z^3 -
38962833 z^4 + 125375523 z^5 - 177681280 z^6 + 133367808 z^7 -
52101120 z^8 + 8388608 z^9) EllipticE[(1/2) (1 - Sqrt[z])] -
2 Sqrt[z] (-185640 + 318240 z + 272493 z^2 + 487305 z^3 - 38962833 z^4 +
125375523 z^5 - 177681280 z^6 + 133367808 z^7 - 52101120 z^8 +
8388608 z^9) EllipticE[(1/2) (1 + Sqrt[z])] +
(371280 - 185640 Sqrt[z] - 914940 z + 318240 z^(3/2) - 109395 z^2 +
272493 z^(5/2) - 515151 z^3 + 487305 z^(7/2) - 5165433 z^4 -
38962833 z^(9/2) + 22767927 z^5 + 125375523 z^(11/2) - 36814960 z^6 -
177681280 z^(13/2) + 29998080 z^7 + 133367808 z^(15/2) - 12435456 z^8 -
52101120 z^(17/2) + 2097152 z^9 + 8388608 z^(19/2))
EllipticK[(1/2) (1 - Sqrt[z])] + (-371280 - 185640 Sqrt[z] + 914940 z +
318240 z^(3/2) + 109395 z^2 + 272493 z^(5/2) + 515151 z^3 +
487305 z^(7/2) + 5165433 z^4 - 38962833 z^(9/2) - 22767927 z^5 +
125375523 z^(11/2) + 36814960 z^6 - 177681280 z^(13/2) - 29998080 z^7 +
133367808 z^(15/2) + 12435456 z^8 - 52101120 z^(17/2) - 2097152 z^9 +
8388608 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 21 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 15 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["21", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["15", "4"], HypergeometricPFQ, Rule[Editable, True], 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HypergeometricPFQ[{},{},z] | HypergeometricPFQ[{},{b},z] | HypergeometricPFQ[{a},{},z] | HypergeometricPFQ[{a},{b},z] | HypergeometricPFQ[{a1},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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