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http://functions.wolfram.com/07.31.03.0136.01
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HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3],
Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]},
1] == (-1)^(-Subscript[b, 1] - 1) Sqrt[Gamma[1 - Subscript[a, 1]]]
Sqrt[Gamma[1 - Subscript[a, 2]]] Sqrt[Gamma[1 - Subscript[a, 3]]]
Sqrt[Gamma[1 - Subscript[a, 4]]]
Sqrt[Gamma[1 + Subscript[a, 1] - Subscript[b, 1]]]
Sqrt[Gamma[1 + Subscript[a, 2] - Subscript[b, 1]]]
Sqrt[Gamma[1 + Subscript[a, 3] - Subscript[b, 1]]]
Sqrt[Gamma[1 + Subscript[a, 4] - Subscript[b, 1]]] Gamma[Subscript[b, 2]]
Gamma[Subscript[b, 3]]
(SixJSymbol[{(-1 + Subscript[b, 3] - Subscript[a, 1] - Subscript[a, 4])/2,
(-1 + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 3])/2,
(-1 - Subscript[b, 1] + Subscript[a, 1] + Subscript[a, 2])/2},
{(-1 - Subscript[a, 2] - Subscript[a, 3] + Subscript[b, 3])/2,
(-1 - Subscript[a, 2] - Subscript[a, 4] + Subscript[b, 2])/2,
(-1 + Subscript[a, 3] + Subscript[a, 4] - Subscript[b, 1])/2}]/
(Gamma[1 - Subscript[b, 1]]
Sqrt[Gamma[-Subscript[a, 1] + Subscript[b, 3]]]
Sqrt[Gamma[-Subscript[a, 2] + Subscript[b, 2]]]
Sqrt[Gamma[-Subscript[a, 3] + Subscript[b, 2]]]
Sqrt[Gamma[-Subscript[a, 4] + Subscript[b, 2]]]
Sqrt[Gamma[Subscript[b, 2] - Subscript[a, 1]]]
Sqrt[Gamma[-Subscript[a, 2] + Subscript[b, 3]]]
Sqrt[Gamma[-Subscript[a, 3] + Subscript[b, 3]]]
Sqrt[Gamma[-Subscript[a, 4] + Subscript[b, 3]]])) /;
Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4] -
Subscript[b, 1] - Subscript[b, 2] - Subscript[b, 3] == -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", SubscriptBox["b", "1"]]], "-", "1"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "2"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "3"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "4"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"]]], "]"]]], SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "4"], "-", SubscriptBox["b", "1"]]], "]"]]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "3"], "]"]], " ", RowBox[List[RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["b", "3"], "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "4"]]], "2"], ",", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], "2"], ",", FractionBox[RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["b", "1"], "+", SubscriptBox["a", "1"], "+", SubscriptBox["a", "2"]]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", SubscriptBox["b", "3"]]], "2"], ",", FractionBox[RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["a", "2"], "-", SubscriptBox["a", "4"], "+", SubscriptBox["b", "2"]]], "2"], ",", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", SubscriptBox["a", "3"], "+", SubscriptBox["a", "4"], "-", SubscriptBox["b", "1"]]], "2"]]], "}"]]]], "]"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["b", "1"]]], "]"]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["b", "3"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["b", "2"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "3"]]], "+", SubscriptBox["b", "2"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "4"]]], "+", SubscriptBox["b", "2"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["b", "3"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "3"]]], "+", SubscriptBox["b", "3"]]], "]"]]], " ", SqrtBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", SubscriptBox["a", "4"]]], "+", SubscriptBox["b", "3"]]], "]"]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["a", "2"], "+", SubscriptBox["a", "3"], "+", SubscriptBox["a", "4"], "-", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"], "-", SubscriptBox["b", "3"]]], "\[Equal]", RowBox[List["-", "1"]]]]]]]]
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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> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["{", Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> <mtext>   </mtext> <mtable> <mtr> <mtd> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mtd> <mtd> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mtd> <mtd> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mtd> <mtd> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mtd> <mtd> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> + </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mtd> </mtr> </mtable> <mtext>   </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox["}", Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> 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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},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] | |
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