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http://functions.wolfram.com/07.23.03.agwz.01
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Hypergeometric2F1[-(3/4), 17/4, -(9/2), z] ==
(1/(18720 Pi^(3/2)))
(((1/(-1 + z)^8) (2 Sqrt[z] (9360 - 64740 z + 184353 z^2 - 263484 z^3 +
150150 z^4 - 582120 z^5 + 437745 z^6 - 155936 z^7 + 22528 z^8)
EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^8)
(2 Sqrt[z] (9360 - 64740 z + 184353 z^2 - 263484 z^3 + 150150 z^4 -
582120 z^5 + 437745 z^6 - 155936 z^7 + 22528 z^8)
EllipticE[(1/2) (1 + Sqrt[z])]) +
(1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7))
((18720 - 28080 Sqrt[z] - 115440 z + 180180 z^(3/2) + 283530 z^2 -
467883 z^(5/2) - 322101 z^3 + 585585 z^(7/2) + 75075 z^4 -
225225 z^(9/2) + 345345 z^5 + 236775 z^(11/2) - 336105 z^6 -
101640 z^(13/2) + 139040 z^7 + 16896 z^(15/2) - 22528 z^8)
EllipticK[(1/2) (1 - Sqrt[z])]) +
(1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8))
((-18720 - 28080 Sqrt[z] + 115440 z + 180180 z^(3/2) - 283530 z^2 -
467883 z^(5/2) + 322101 z^3 + 585585 z^(7/2) - 75075 z^4 -
225225 z^(9/2) - 345345 z^5 + 236775 z^(11/2) + 336105 z^6 -
101640 z^(13/2) - 139040 z^7 + 16896 z^(15/2) + 22528 z^8)
EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^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> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mn> 17 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 9 </mn> <mn> 2 </mn> </mfrac> </mrow> <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["3", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["17", "4"], HypergeometricPFQ, 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<mn> 9360 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 22528 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 155936 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 437745 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 582120 </mn> <mo> ⁢ </mo> <msup> 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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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