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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.03.0024.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, 1] == ((Gamma[Subscript[\[Psi], q]] Product[Gamma[Subscript[b, k]], {k, 1, q}])/ Product[Gamma[Subscript[a, k]], {k, 3, q + 1}]) Sum[(Pochhammer[Subscript[\[Psi], q], k] HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, k])/(Gamma[Subscript[\[Psi], q] + Subscript[a, 1] + k] Gamma[Subscript[\[Psi], q] + Subscript[a, 2] + k]), {k, 0, Infinity}] /; Subscript[\[Psi], q] == Sum[Subscript[b, j], {j, 1, q}] - Sum[Subscript[a, j], {j, 1, q + 1}] && Re[Subscript[\[Psi], q]] > 0 && Re[Subscript[a, 3]] > 0 && \[Ellipsis] && Re[Subscript[a, q + 1]] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["\[Psi]", "q"], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "3"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Psi]", "q"], ",", "k"]], "]"]], RowBox[List["HypergeometricPFQExpansionCoefficient", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "q"], "+", SubscriptBox["a", "1"], "+", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Psi]", "q"], "+", SubscriptBox["a", "2"], "+", "k"]], "]"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "q"], "\[Equal]", RowBox[List[RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "q"], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["b", "j"]]], "-", RowBox[List[StyleBox[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "+", "1"]]], Rule[LimitsPositioningTokens, List["\[Sum]", "\[Product]", "\[Intersection]", "\[Union]", "\[UnionPlus]", "\[Wedge]", "\[Vee]", "lim", "max", "min", "\[CirclePlus]", "\[CircleMinus]", "\[CircleTimes]", "\[CircleDot]"]]], SubscriptBox["a", "j"]]]]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Psi]", "q"], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "3"], "]"]], ">", "0"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "]"]], ">", "0"]]]]]]]]

 MathML Form

 q + 1 F q ( a 1 , , a q + 1 ; b 1 , , b q ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] Γ ( ψ q ) k = 1 q Γ ( b k ) k = 3 q + 1 Γ ( a k ) k = 0 ( ψ q ) k k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) Γ ( k + a 1 + ψ q ) Γ ( k + a 2 + ψ q ) TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "q"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "q", ")"]]], "(", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]]]], ")"]]]], RowBox[List[RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; ψ q j = 1 q b j - j = 1 q + 1 a j Re ( ψ q ) > 0 Re ( a 3 ) > 0 Re ( a q + 1 ) > 0 q + 1 F q ( a 1 , , a q + 1 ; b 1 , , b q ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] Γ ( ψ q ) k = 1 q Γ ( b k ) k = 3 q + 1 Γ ( a k ) k = 0 ( ψ q ) k k ( q ) ( { a 1 , , a q + 1 } , { b 1 , , b q } ) Γ ( k + a 1 + ψ q ) Γ ( k + a 2 + ψ q ) TagBox[TagBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Psi]", "q"], ")"]], "k"], Pochhammer], " ", RowBox[List[SubsuperscriptBox["\[ScriptCapitalE]", "k", RowBox[List["(", "q", ")"]]], "(", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]]]], ")"]]]], RowBox[List[RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]], " ", RowBox[List["\[CapitalGamma]", "(", RowBox[List["k", "+", SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "q"]]], ")"]]]]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; ψ q j = 1 q b j - j = 1 q + 1 a j Re ( ψ q ) > 0 Re ( a 3 ) > 0 Re ( a q + 1 ) > 0 [/itex]

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

 2001-10-29