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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0028.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 1], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]])) (1 + O[1/z]))/ (-z)^Subscript[a, 3] + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[a, 3] - Subscript[a, 1]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 3]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) (Log[-z] - 2 EulerGamma - PolyGamma[Subscript[a, 1]] + PolyGamma[Subscript[a, 3] - Subscript[a, 1]] - PolyGamma[Subscript[b, 1] - Subscript[a, 1]] - PolyGamma[Subscript[b, 2] - Subscript[a, 1]]) (1 + O[1/z]))/ (-z)^Subscript[a, 1] /; (Abs[z] -> Infinity) && !Element[Subscript[a, 1] - Subscript[a, 3], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "1"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "3"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "3"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "1"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["2", "EulerGamma"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "1"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 3 F 2 ( a 1 , a 1 , a 3 ; b 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], 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]]]], 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 ( a 3 - a 1 ) Γ ( a 1 ) Γ ( a 3 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( a 3 - a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( b 1 - a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( b 2 - a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a 1 ) - 2 TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ( - z ) - a 1 ( 1 + O ( 1 z ) ) + Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 - a 3 ) Γ ( a 2 - a 3 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( b 1 - a 3 ) Γ ( b 2 - a 3 ) ( - z ) - a 3 ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) a 1 - a 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Proportional HypergeometricPFQ Subscript a 1 Subscript a 1 Subscript a 3 Subscript b 1 Subscript b 2 z Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript a 3 -1 Subscript a 1 Gamma Subscript a 1 Gamma Subscript a 3 Gamma Subscript b 1 -1 Subscript a 1 Gamma Subscript b 2 -1 Subscript a 1 -1 -1 z PolyGamma Subscript a 3 -1 Subscript a 1 -1 PolyGamma Subscript b 1 -1 Subscript a 1 -1 PolyGamma Subscript b 2 -1 Subscript a 1 -1 PolyGamma Subscript a 1 -1 2 -1 z -1 Subscript a 1 1 O 1 z -1 Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript a 1 -1 Subscript a 3 Gamma Subscript a 2 -1 Subscript a 3 Gamma Subscript a 1 Gamma Subscript a 2 Gamma Subscript b 1 -1 Subscript a 3 Gamma Subscript b 2 -1 Subscript a 3 -1 -1 z -1 Subscript a 3 1 O 1 z -1 Rule z Subscript a 1 -1 Subscript a 3 [/itex]

 Rule Form

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

 2001-10-29