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http://functions.wolfram.com/07.23.03.0053.01
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Hypergeometric2F1[1/2, b, 5/2 - 2 b, 1/4] ==
((2^(2 b) Sqrt[Pi])/3) (Gamma[5/2 - 2 b]/Gamma[3/2 - b]^2)
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Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", "b", ",", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", "b"]]]], ",", FractionBox["1", "4"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", "b"]]], SqrtBox["\[Pi]"]]], "3"], FractionBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", "b"]]]], "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "b"]], "]"]], "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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", " ", "b"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", "4"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> ⩵ </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> b </ci> <apply> <plus /> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", "b_", ",", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", " ", "b_"]]]], ",", FractionBox["1", "4"]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "b"]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["5", "2"], "-", RowBox[List["2", " ", "b"]]]], "]"]]]], RowBox[List["3", " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "b"]], "]"]], "2"]]]]]]]] |
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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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