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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.06.0041.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] \[Proportional] Sum[(Subscript[c, k] (1 + O[1/z]))/(-z)^Subscript[a, k], {k, 1, p}] + KroneckerDelta[q, p + 1] Subscript[e, 1] (-z)^\[Chi] Cos[2 Sqrt[-z] + \[Chi] Pi] (1 + O[1/Sqrt[-z]]) + (UnitStep[q - p] - KroneckerDelta[q, p + 1]) Subscript[e, 2] z^\[Chi] Exp[\[Beta] z^(1/\[Beta])] (1 + O[1/z^(1/\[Beta])]) /; (Abs[z] -> Infinity) && \[Beta] == q - p + 1 && \[Chi] == (1/\[Beta]) ((\[Beta] - 1)/2 + Sum[Subscript[a, k], {k, 1, p}] - Sum[Subscript[b, k], {k, 1, q}]) && Subscript[c, k] == (Gamma[Subscript[a, k]] Product[Gamma[Subscript[b, j]], {j, 1, q}] Product[If[j == k, 1, Gamma[Subscript[a, j] - Subscript[a, k]]], {j, 1, p}])/(Product[Gamma[Subscript[a, j]], {j, 1, p}] Product[Gamma[Subscript[b, j] - Subscript[a, k]], {j, 1, q}]) && 2 Subscript[e, 2] == Subscript[e, 1] == ((2 (2 Pi)^((1 - \[Beta])/2))/Sqrt[\[Beta]]) (Product[Gamma[Subscript[b, k]], {k, 1, q}]/ Product[Gamma[Subscript[a, k]], {k, 1, p}]) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= p && 1 <= k <= p, Subscript[a, j] != Subscript[a, k]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[SubscriptBox["c", "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[RowBox[List["KroneckerDelta", "[", RowBox[List["q", ",", RowBox[List["p", "+", "1"]]]], "]"]], SubscriptBox["e", "1"], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], RowBox[List["Cos", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Chi]", " ", "\[Pi]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["q", "-", "p"]], "]"]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["q", ",", RowBox[List["p", "+", "1"]]]], "]"]]]], ")"]], SubscriptBox["e", "2"], " ", SuperscriptBox["z", "\[Chi]"], " ", RowBox[List["Exp", "[", RowBox[List["\[Beta]", " ", SuperscriptBox["z", FractionBox["1", "\[Beta]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["1", "/", "\[Beta]"]]]], "]"]]]], ")"]]]]]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Beta]", "\[Equal]", RowBox[List["q", "-", "p", "+", "1"]]]], "\[And]", RowBox[List["\[Chi]", "\[Equal]", RowBox[List[FractionBox["1", "\[Beta]"], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Beta]", "-", "1"]], "2"], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]]]], ")"]]]]]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "j"], "]"]]]], ")"]], 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["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]]]], "\[And]", RowBox[List[RowBox[List["2", SubscriptBox["e", "2"]]], "\[Equal]", SubscriptBox["e", "1"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], FractionBox[RowBox[List["1", "-", "\[Beta]"]], "2"]], " "]], SqrtBox["\[Beta]"]], FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]]]]]], "\[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["(", RowBox[List[SubscriptBox["a", "j"], "\[NotEqual]", SubscriptBox["a", "k"]]], ")"]]]]]]]]]]

 MathML Form

 p F q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", 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", "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", "q"], 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] k = 1 p c k ( - z ) - a k ( 1 + O ( 1 z ) ) + δ KroneckerDelta q , p + 1 e 1 ( - z ) χ cos ( π χ + 2 - z ) ( 1 + O ( 1 - z ) ) + ( θ UnitStep ( q - p ) - δ KroneckerDelta q , p + 1 ) e 2 z χ β z 1 / β ( 1 + O ( 1 z 1 / β ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) β q - p + 1 χ 1 β ( β - 1 2 + k = 1 p a k - k = 1 q b k ) c k Γ ( a k ) ( j = 1 q Γ ( b j ) ) j = 1 j k p Γ ( a j - a k ) ( j = 1 p Γ ( a j ) ) j = 1 q Γ ( b j - a k ) 2 e 2 e 1 2 ( 2 π ) 1 - β 2 k = 1 q Γ ( b k ) β k = 1 p Γ ( a k ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j p 1 k p ( a j a k ) FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox F FormBox q 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 q 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 UnderoverscriptBox RowBox k = 1 p RowBox SubscriptBox c k SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a k RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) + RowBox SubscriptBox InterpretationBox δ KroneckerDelta Rule Editable Rule Selectable RowBox q , RowBox p + 1 SubscriptBox e 1 SuperscriptBox RowBox ( RowBox - z ) χ RowBox cos ( RowBox RowBox π χ + RowBox 2 SqrtBox RowBox - z ) RowBox ( RowBox 1 + RowBox O RowBox ( FractionBox 1 SqrtBox RowBox - z ) ) + RowBox RowBox ( RowBox RowBox InterpretationBox θ UnitStep Rule Editable Rule Selectable RowBox ( RowBox q - p ) - SubscriptBox InterpretationBox δ KroneckerDelta Rule Editable Rule Selectable RowBox q , RowBox p + 1 ) SubscriptBox e 2 SuperscriptBox z χ SuperscriptBox RowBox β SuperscriptBox z RowBox 1 / β RowBox ( RowBox 1 + RowBox O ( FractionBox 1 SuperscriptBox z RowBox 1 / β ) ) /; RowBox RowBox ( RowBox RowBox z ) RowBox β RowBox q - p + 1 RowBox χ RowBox FractionBox 1 β RowBox ( RowBox FractionBox RowBox β - 1 2 + RowBox UnderoverscriptBox RowBox k = 1 p SubscriptBox a k - RowBox UnderoverscriptBox RowBox k = 1 q SubscriptBox b k ) RowBox SubscriptBox c k FractionBox RowBox RowBox Γ ( SubscriptBox a k ) RowBox ( RowBox UnderoverscriptBox RowBox j = 1 q RowBox Γ ( SubscriptBox b j ) ) RowBox ErrorBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k p RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox a k ) RowBox RowBox ( RowBox UnderoverscriptBox RowBox j = 1 p RowBox Γ ( SubscriptBox a j ) ) RowBox UnderoverscriptBox RowBox j = 1 q RowBox Γ ( RowBox SubscriptBox b j - SubscriptBox a k ) RowBox RowBox 2 SubscriptBox e 2 SubscriptBox e 1 FractionBox RowBox 2 SuperscriptBox RowBox ( RowBox 2 π ) FractionBox RowBox 1 - β 2 RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( SubscriptBox b k ) RowBox SqrtBox β RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) 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 SubscriptBox a j SubscriptBox a k ) TraditionalForm [/itex]

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

 2001-10-29

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