html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.31.06.0033.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] \[Proportional] (Product[Gamma[Subscript[b, j]], {j, 1, q}]/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, q}]) (1 + O[1/z]))/(-z)^Subscript[a, k], {k, 1, p}] + (Product[Gamma[Subscript[b, j]], {j, 1, q}]/ Product[Gamma[Subscript[a, k]], {k, 1, p}]) ((2 Pi)^((1 - \[Beta])/2)/ Sqrt[\[Beta]]) Exp[\[Beta] z^(1/\[Beta])] z^\[Chi] (1 + O[1/z^(1/\[Beta])]) /; q - p >= 2 && (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}]) && 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["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[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "j"], "]"]]]], 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"]], "q"], 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[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "j"], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], FractionBox[RowBox[List["1", "-", "\[Beta]"]], "2"]], " "]], SqrtBox["\[Beta]"]], RowBox[List["Exp", "[", RowBox[List["\[Beta]", " ", SuperscriptBox["z", FractionBox["1", "\[Beta]"]]]], "]"]], SuperscriptBox["z", "\[Chi]"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["1", "/", "\[Beta]"]]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["q", "-", "p"]], "\[GreaterEqual]", "2"]], "\[And]", 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["\[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["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]]]]]]

 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] ( 2 π ) 1 - β 2 j = 1 q Γ ( b j ) β k = 1 p Γ ( a k ) z χ exp ( β z 1 / β ) ( 1 + O ( 1 z 1 / β ) ) + j = 1 q Γ ( b j ) k = 1 p Γ ( a k ) k = 1 p Γ ( a k ) j = 1 j k p Γ ( a j - a k ) j = 1 q Γ ( b j - a k ) ( - z ) - a k ( 1 + O ( 1 z ) ) /; q - p 2 ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) β q - p + 1 χ 1 β ( β - 1 2 + k = 1 p a k - k = 1 q b k ) { 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 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 FractionBox RowBox SuperscriptBox RowBox ( RowBox 2 π ) FractionBox RowBox 1 - β 2 RowBox UnderoverscriptBox RowBox j = 1 q RowBox Γ ( SubscriptBox b j ) RowBox SqrtBox β RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) SuperscriptBox z χ RowBox exp ( RowBox β SuperscriptBox z RowBox 1 / β ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 SuperscriptBox z RowBox 1 / β ) ) + RowBox FractionBox RowBox UnderoverscriptBox RowBox j = 1 q RowBox Γ ( SubscriptBox b j ) 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 q 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 q - p 2 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 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]

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

 2001-10-29