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http://functions.wolfram.com/07.24.26.0023.01
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(1 + z)^(1 - a) Hypergeometric2F1Regularized[a, 1 - a, c, z/(1 + z)] ==
(2/(Gamma[-1 + a] Gamma[-1 + a + c]))
MeijerG[{{2 - a, 3 - a - c, 1 - (a + c)/2}, {}},
{{0}, {1 - c, 2 - (a + c)/2}}, z] /;
!IntervalMemberQ[Interval[{-Infinity, -1}], z]
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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> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> c </mi> <annotation encoding='Mathematica'> TagBox[TagBox["c", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]] </annotation> </semantics> <mo> ; 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</mo> <mrow> <mi> z </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> a </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms>  </ms> </apply> <apply> <ci> FormBox </ci> <ms> 2 </ms> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> FormBox </ci> <ms> 1 </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> a </ms> <ci> Hypergeometric2F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> a </ms> </list> </apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> c </ms> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> c </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> G </ms> <ci> MeijerG </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> , </ms> <ms> 3 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <ms> 3 </ms> </list> </apply> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> z </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ❘ </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> - </ms> <ms> a </ms> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> - </ms> <ms> a </ms> <ms> - </ms> <ms> c </ms> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> c </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> 0 </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> c </ms> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> c </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ∉ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> ∞ </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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HypergeometricPFQRegularized[{},{b},z] | HypergeometricPFQRegularized[{a1},{b1},z] | HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] | |
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