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http://functions.wolfram.com/07.23.03.b2jj.01
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Hypergeometric2F1[3, 5, -(29/5), z] ==
-((1/(17218750 (-1 + z)^13)) (17218750 - 268375000 z + 2033296875 z^2 -
10187062500 z^3 + 39639140625 z^4 - 141797250000 z^5 +
682669640625 z^6 + 2068942485684 z^7 + 690875065935 z^8 +
14198498600 z^9 - 334870250 z^10)) - (1/(78125 (1 - z)^(69/5)))
(6432426 Sqrt[2 (5 + Sqrt[5])] z^(34/5) (429 + 440 z + 75 z^2)
ArcTan[1 - ((1 - Sqrt[5]) z^(1/5))/(4 (1 - z)^(1/5)),
-((Sqrt[5/8 + Sqrt[5]/8] z^(1/5))/(1 - z)^(1/5))]) -
(1/(78125 (1 - z)^(69/5))) (6432426 Sqrt[10 - 2 Sqrt[5]] z^(34/5)
(429 + 440 z + 75 z^2) ArcTan[1 - ((1 + Sqrt[5]) z^(1/5))/
(4 (1 - z)^(1/5)), -((Sqrt[5/8 - Sqrt[5]/8] z^(1/5))/
(1 - z)^(1/5))]) + (12864852 z^(34/5) (429 + 440 z + 75 z^2)
Log[1 + z^(1/5)/(1 - z)^(1/5)])/(78125 (1 - z)^(69/5)) +
(3216213 (-1 + Sqrt[5]) z^(34/5) (429 + 440 z + 75 z^2)
Log[1 - ((1 - Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) +
z^(2/5)/(1 - z)^(2/5)])/(78125 (1 - z)^(69/5)) -
(3216213 (1 + Sqrt[5]) z^(34/5) (429 + 440 z + 75 z^2)
Log[1 - ((1 + Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) +
z^(2/5)/(1 - z)^(2/5)])/(78125 (1 - z)^(69/5))
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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> <mn> 3 </mn> <mo> , </mo> <mn> 5 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 29 </mn> <mn> 5 </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["3", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["5", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], 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+ </mo> <mn> 429 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mn> 5 </mn> <mn> 8 </mn> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mn> 5 </mn> </msqrt> <mn> 8 </mn> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 34 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> <mrow> <mn> 78125 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 69 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 6432426 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 10 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 75 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 440 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 429 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mn> 5 </mn> <mn> 8 </mn> </mfrac> <mo> - </mo> <mfrac> <msqrt> <mn> 5 </mn> </msqrt> <mn> 8 </mn> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 34 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> <mrow> <mn> 78125 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 69 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 12864852 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 75 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 440 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 429 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 34 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> <mrow> <mn> 78125 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 69 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 3216213 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( 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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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