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 HypergeometricPFQ

 http://functions.wolfram.com/07.26.07.0004.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == (1/(2 Pi I)) ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]])) Integrate[(Gamma[s] Gamma[Subscript[a, 1] - s] Gamma[Subscript[a, 2] - s])/ (Gamma[Subscript[b, 1] - s] Gamma[Subscript[b, 2] - s] Gamma[Subscript[b, 3] - s])/(-z)^s, {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; 0 < \[Gamma] < Min[Re[Subscript[a, 1]], Re[Subscript[a, 2]], 1/4 + Re[Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] - Subscript[a, 1] - Subscript[a, 2]]/2] && z < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "3"], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "1"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "2"], "-", "s"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "3"], "-", "s"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]", "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", RowBox[List["Re", "[", SubscriptBox["a", "2"], "]"]], ",", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Re", "[", RowBox[List[SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"], "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], "]"]], "2"]]]]], "]"]]]], "\[And]", RowBox[List["z", "<", "0"]]]]]]]]

 MathML Form

 2 F 3 ( a 1 , a 2 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", 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]]]], 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["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) ( 2 π ) Γ ( a 1 ) Γ ( a 2 ) γ - γ + Γ ( s ) Γ ( a 1 - s ) Γ ( a 2 - s ) Γ ( b 1 - s ) Γ ( b 2 - s ) Γ ( b 3 - s ) ( - z ) - s s /; 0 < γ < min ( Re ( a 1 ) , Re ( a 2 ) , 1 2 Re ( b 1 + b 2 + b 3 - a 1 - a 2 ) + 1 4 ) z < 0 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 Subscript b 3 z Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 2 Gamma Subscript a 1 Gamma Subscript a 2 -1 s γ -1 γ Gamma s Gamma Subscript a 1 -1 s Gamma Subscript a 2 -1 s Gamma Subscript b 1 -1 s Gamma Subscript b 2 -1 s Gamma Subscript b 3 -1 s -1 -1 z -1 s 0 γ Subscript a 1 Subscript a 2 1 2 Subscript b 1 Subscript b 2 Subscript b 3 -1 Subscript a 1 -1 Subscript a 2 1 4 z 0 [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2001-10-29