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http://functions.wolfram.com/07.23.03.0669.01
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Hypergeometric2F1[a, -(5/6) + (5 a)/3, 2 a, 4/GoldenRatio^3] ==
3^a 5^(1/6 - (5 a)/6) GoldenRatio^(-3 + 5 a) Gamma[1/6 + a/3]
Gamma[1/2 + a/3] (GoldenRatio/(Gamma[7/30 + a/3] Gamma[13/30 + a/3]) -
1/(Gamma[1/30 + a/3] Gamma[19/30 + a/3]))
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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> <mi> a </mi> <mo> , </mo> <mrow> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> ; </mo> <mfrac> <mn> 4 </mn> <msup> <mi> ϕ </mi> <mn> 3 </mn> </msup> </mfrac> </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["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["5", " ", "a"]], "3"], "-", FractionBox["5", "6"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", " ", "a"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["4", SuperscriptBox[TagBox["\[Phi]", Function[List[], GoldenRatio]], "3"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo>  </mo> <mrow> <msup> <mn> 3 </mn> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mn> 5 </mn> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mn> 6 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 7 </mn> <mn> 30 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 13 </mn> <mn> 30 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 30 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 19 </mn> <mn> 30 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <ci> GoldenRatio </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <power /> <cn type='integer'> 5 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> a </ci> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> GoldenRatio </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> a </ci> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> GoldenRatio </ci> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 7 <sep /> 30 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 13 <sep /> 30 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 30 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 19 <sep /> 30 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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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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