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http://functions.wolfram.com/07.32.03.0136.01
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HypergeometricPFQRegularized[{-n, b, c, d, e, b/2 + 1,
2 b - c - d - e + n + 1}, {b/2, 1 + b - c, 1 + b - d, 1 + b - e,
1 + b + n, c + d + e - b - n}, 1] == (-Gamma[1 + b - c - d - e])
Pochhammer[1 + b - c - d, n] Pochhammer[1 + b - c - e, n]
Pochhammer[1 + b - d - e, n] (Sin[(b - c - d - e + n) Pi]/
(Pi Gamma[b/2] Gamma[1 + b] Gamma[1 + b - c + n] Gamma[1 + b - d + n]
Gamma[1 + b - e + n])) /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", "b", ",", "c", ",", "d", ",", "e", ",", RowBox[List[FractionBox["b", "2"], "+", "1"]], ",", RowBox[List[RowBox[List["2", "b"]], "-", "c", "-", "d", "-", "e", "+", "n", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["b", "2"], ",", RowBox[List["1", "+", "b", "-", "c"]], ",", RowBox[List["1", "+", "b", "-", "d"]], ",", RowBox[List["1", "+", "b", "-", "e"]], ",", RowBox[List["1", "+", "b", "+", "n"]], ",", RowBox[List["c", "+", "d", "+", "e", "-", "b", "-", "n"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c", "-", "d", "-", "e"]], "]"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "b", "-", "c", "-", "d"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "b", "-", "c", "-", "e"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "b", "-", "d", "-", "e"]], ",", "n"]], "]"]], " ", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c", "-", "d", "-", "e", "+", "n"]], ")"]], " ", "\[Pi]"]], "]"]], "/", RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", FractionBox["b", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c", "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "d", "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "e", "+", "n"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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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> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 7 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 6 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mi> e </mi> <mo> , </mo> <mrow> <mfrac> <mi> b </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mi> b </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> d </mi> <mo> + </mo> <mi> e </mi> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["7", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["6", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["c", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["d", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["e", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "b"]], "-", "c", "-", "d", "-", "e", "+", "n", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["b", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "d", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "e", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "n", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "b", "+", "d", "+", "e", "-", "n"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", "d", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", "e", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> d </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "d", "-", "e", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> d </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> b </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> b </ci> <ci> c </ci> <ci> d </ci> <ci> e </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <times /> <ci> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> d </ci> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </list> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <ci> n </ci> </apply> <pi /> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
