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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0030.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, p + 1]}, z] \[Proportional] (1/Product[Gamma[Subscript[a, k]], {k, 1, p}]) Sum[(((Gamma[Subscript[a, k]] Product[If[j == k, 1, Gamma[Subscript[a, j] - Subscript[a, k]]], {j, 1, p}])/ Product[Gamma[Subscript[b, j] - Subscript[a, k]], {j, 1, p + 1}]) (1 + O[1/z]))/(-z)^Subscript[a, k], {k, 1, p}] + (1/Product[Gamma[Subscript[a, k]], {k, 1, p}]) (1/(2 Sqrt[Pi])) (-z)^\[Chi] (E^(I (\[Chi] Pi + 2 Sqrt[-z])) (1 + O[1/Sqrt[-z]]) + (1 + O[1/Sqrt[-z]])/E^(I (\[Chi] Pi + 2 Sqrt[-z]))) /; (Abs[z] -> Infinity) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= p && 1 <= k <= p, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["p", "+", "1"]]]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]], "]"]]]]]], RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["p", "+", "1"]]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "k"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], FractionBox["1", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Chi]", " ", "\[Pi]"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Chi]", " ", "\[Pi]"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "p"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "p"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]]]]]]

 MathML Form

 p F ~ p + 1 ( a 1 , , a p ; b 1 , , b p + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["p", "+", "1"]], 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", "p"], 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", RowBox[List["p", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] 1 2 π k = 1 p Γ ( a k ) ( - z ) χ ( ( π χ + 2 - z ) ( 1 + O ( 1 - z ) ) + - ( π χ + 2 - z ) ( 1 + O ( 1 - z ) ) ) + 1 k = 1 p Γ ( a k ) k = 1 p Γ ( a k ) j = 1 j k p Γ ( a j - a k ) j = 1 p + 1 Γ ( b j - a k ) ( - z ) - a k ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j p 1 k p ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ FormBox RowBox p + 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox a p HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox b RowBox p + 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ RowBox RowBox FractionBox 1 RowBox 2 SqrtBox π RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) SuperscriptBox RowBox ( RowBox - z ) χ RowBox ( RowBox RowBox SuperscriptBox RowBox RowBox ( RowBox RowBox π χ + RowBox 2 SqrtBox RowBox - z ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 SqrtBox RowBox - z ) ) + RowBox SuperscriptBox RowBox RowBox - RowBox ( RowBox RowBox π χ + RowBox 2 SqrtBox RowBox - z ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 SqrtBox RowBox - z ) ) ) + RowBox FractionBox 1 RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox RowBox k = 1 p ErrorBox RowBox FractionBox RowBox RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k p RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox a k ) RowBox UnderoverscriptBox RowBox j = 1 RowBox p + 1 RowBox Γ ( RowBox SubscriptBox b j - SubscriptBox a k ) SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a k RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) /; RowBox RowBox ( RowBox RowBox z ) RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 1 j p RowBox 1 k p RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", RowBox[List["p_", "+", "1"]]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "k"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["p", "+", "1"]]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Chi]", " ", "\[Pi]"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List[RowBox[List["-", "z"]], ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", FractionBox["1", "2"]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Chi]", " ", "\[Pi]"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List[RowBox[List["-", "z"]], ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", FractionBox["1", "2"]]], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", SqrtBox["\[Pi]"]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[NotEqual]", "k"]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "p"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "p"]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]]]], ")"]]]]]]]]]]]]

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

 2001-10-29