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; }

 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0024.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] \[Proportional] (1/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) ((((Gamma[Subscript[a, 1]] Product[Gamma[Subscript[a, k] - Subscript[a, 1]], {k, 2, q + 1}])/ Product[Gamma[Subscript[b, k] - Subscript[a, 1]], {k, 1, q}]) (1 + O[1/z]))/(-z)^Subscript[a, 1] + ((((-1)^(Subscript[a, 2] - Subscript[a, 1]) Gamma[Subscript[a, 2]] Product[Gamma[Subscript[a, k] - Subscript[a, 2]], {k, 3, q + 1}])/ ((Subscript[a, 2] - Subscript[a, 1])! Product[Gamma[Subscript[b, k] - Subscript[a, 2]], {k, 1, q}])) (-EulerGamma + PolyGamma[1 + Subscript[a, 2] - Subscript[a, 1]] - PolyGamma[Subscript[a, 2]] + Sum[PolyGamma[Subscript[a, k] - Subscript[a, 2]], {k, 3, q + 1}] - Sum[PolyGamma[Subscript[b, k] - Subscript[a, 2]], {k, 1, q}] + Log[-z]) (1 + O[1/z]))/(-z)^Subscript[a, 2] + ((((-1)^(Subscript[a, 2] - Subscript[a, 1]) Gamma[Subscript[a, 3]] Product[Gamma[Subscript[a, k] - Subscript[a, 3]], {k, 4, q + 1}])/ (2 (Subscript[a, 3] - Subscript[a, 2])! (Subscript[a, 3] - Subscript[a, 1])! Product[Gamma[Subscript[b, k] - Subscript[a, 3]], {k, 1, q}])) ((-EulerGamma + PolyGamma[1 + Subscript[a, 3] - Subscript[a, 2]] + PolyGamma[1 + Subscript[a, 3] - Subscript[a, 1]] - PolyGamma[Subscript[a, 3]] + Sum[PolyGamma[Subscript[a, k] - Subscript[a, 3]], {k, 4, q + 1}] - Sum[PolyGamma[Subscript[b, k] - Subscript[a, 3]], {k, 1, q}] + Log[-z])^2 + ((5 Pi^2)/6 - PolyGamma[1, 1 + Subscript[a, 3] - Subscript[a, 2]] - PolyGamma[1, 1 + Subscript[a, 3] - Subscript[a, 1]] + PolyGamma[1, Subscript[a, 3]] + Sum[PolyGamma[1, Subscript[a, k] - Subscript[a, 3]], {k, 4, q + 1}] - Sum[PolyGamma[1, Subscript[b, k] - Subscript[a, 3]], {k, 1, q}])) (1 + O[1/z]))/(-z)^Subscript[a, 3] + Sum[(((Gamma[Subscript[a, k]] Product[If[j != k, Gamma[Subscript[a, j] - Subscript[a, k]], 1], {j, 1, q + 1}])/ Product[Gamma[Subscript[b, j] - Subscript[a, k]], {j, 1, q}]) (1 + O[1/z]))/(-z)^Subscript[a, k], {k, 4, q + 1}]) /; (Abs[z] -> Infinity) && Element[Subscript[a, 2] - Subscript[a, 1], Integers] && Subscript[a, 2] - Subscript[a, 1] >= 0 && Element[Subscript[a, 3] - Subscript[a, 2], Integers] && Subscript[a, 3] - Subscript[a, 2] >= 0 && !Element[Subscript[a, k] - Subscript[a, 1], Integers] && 4 <= k <= q + 1 && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 4 <= j <= q + 1 && 4 <= k <= q + 1, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

 Standard Form

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RowBox[List["4", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["4", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]], RowBox[List["(", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]]]]]]

 MathML Form

 q + 1 F ~ q ( a 1 , , a q + 1 ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["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["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] 1 k = 1 q + 1 Γ ( a k ) ( Γ ( a 1 ) k = 2 q + 1 Γ ( a k - a 1 ) k = 1 q Γ ( b k - a 1 ) ( - z ) - a 1 ( 1 + O ( 1 z ) ) + ( - 1 ) a 2 - a 1 Γ ( a 2 ) k = 3 q + 1 Γ ( a k - a 2 ) ( a 2 - a 1 ) ! k = 1 q Γ ( b k - a 2 ) ( - z ) - a 2 ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( a 2 - a 1 + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a 2 ) + k = 3 q + 1 ψ TagBox["\[Psi]", PolyGamma] ( a k - a 2 ) - k = 1 q ψ TagBox["\[Psi]", PolyGamma] ( b k - a 2 ) - TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ( 1 + O ( 1 z ) ) + ( - 1 ) a 2 - a 1 Γ ( a 3 ) k = 4 q + 1 Γ ( a k - a 3 ) 2 ( a 3 - a 2 ) ! ( a 3 - a 1 ) ! k = 1 q Γ ( b k - a 3 ) ( - z ) - a 3 ( ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( a 3 - a 1 + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( a 3 - a 2 + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a 3 ) + k = 4 q + 1 ψ TagBox["\[Psi]", PolyGamma] ( a k - a 3 ) - k = 1 q ψ TagBox["\[Psi]", PolyGamma] ( b k - a 3 ) - TagBox["\[DoubledGamma]", Function[EulerGamma]] ) 2 + ( 5 π 2 6 - ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a 3 - a 1 + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a 3 - a 2 + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a 3 ) + k = 4 q + 1 ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a k - a 3 ) - k = 1 q ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( b k - a 3 ) ) ) ( O ( 1 z ) + 1 ) + k = 4 q + 1 Γ ( a k ) j = 1 j k q + 1 Γ ( a j - a k ) j = 1 q Γ ( b j - a k ) ( - z ) - a k ( 1 + O ( 1 z ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) a 2 - a 1 a 3 - a 2 TagBox[RowBox[List["\[DoubleStruckCapitalN]", "\[And]", RowBox[List[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"]]], "\[Element]", TagBox["\[DoubleStruckCapitalN]", Function[Integers]]]]]], Function[Integers]] a k - a 1 4 k q + 1 TagBox[RowBox[List["\[DoubleStruckCapitalZ]", "\[And]", RowBox[List["4", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]], Function[Integers]] { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 4 j q + 1 4 k q + 1 ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox a RowBox q + 1 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 FractionBox 1 RowBox UnderoverscriptBox RowBox k = 1 RowBox q + 1 RowBox Γ ( SubscriptBox a k ) RowBox ( RowBox RowBox FractionBox RowBox RowBox Γ ( SubscriptBox a 1 ) RowBox UnderoverscriptBox RowBox k = 2 RowBox q + 1 RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox a 1 ) RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( RowBox SubscriptBox b k - SubscriptBox a 1 ) SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a 1 RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) + RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox SubscriptBox a 2 - SubscriptBox a 1 RowBox Γ ( SubscriptBox a 2 ) RowBox UnderoverscriptBox RowBox k = 3 RowBox q + 1 RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox a 2 ) RowBox RowBox RowBox ( RowBox SubscriptBox a 2 - SubscriptBox a 1 ) ! RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( RowBox SubscriptBox b k - SubscriptBox a 2 ) SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a 2 RowBox ( RowBox RowBox log ( RowBox - z ) + RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox a 2 - SubscriptBox a 1 + 1 ) - RowBox TagBox ψ PolyGamma ( SubscriptBox a 2 ) + RowBox UnderoverscriptBox RowBox k = 3 RowBox q + 1 RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox a k - SubscriptBox a 2 ) - RowBox UnderoverscriptBox RowBox k = 1 q RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox b k - SubscriptBox a 2 ) - TagBox Function ) RowBox ( RowBox 1 + RowBox O ( FractionBox 1 z ) ) + RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox SubscriptBox a 2 - SubscriptBox a 1 RowBox Γ ( SubscriptBox a 3 ) RowBox UnderoverscriptBox RowBox k = 4 RowBox q + 1 RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox a 3 ) RowBox 2 RowBox RowBox ( RowBox SubscriptBox a 3 - SubscriptBox a 2 ) ! RowBox RowBox ( RowBox SubscriptBox a 3 - SubscriptBox a 1 ) ! RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( RowBox SubscriptBox b k - SubscriptBox a 3 ) SuperscriptBox RowBox ( RowBox - z ) RowBox - SubscriptBox a 3 RowBox ( RowBox SuperscriptBox RowBox ( RowBox RowBox log ( RowBox - z ) + RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox a 3 - SubscriptBox a 1 + 1 ) + RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox a 3 - SubscriptBox a 2 + 1 ) - RowBox TagBox ψ PolyGamma ( SubscriptBox a 3 ) + RowBox UnderoverscriptBox RowBox k = 4 RowBox q + 1 RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox a k - SubscriptBox a 3 ) - RowBox UnderoverscriptBox RowBox k = 1 q RowBox TagBox ψ PolyGamma ( RowBox SubscriptBox b k - SubscriptBox a 3 ) - TagBox Function ) 2 + RowBox ( RowBox FractionBox RowBox 5 SuperscriptBox π 2 6 - RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( 1 ) ( RowBox SubscriptBox a 3 - SubscriptBox a 1 + 1 ) - RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( 1 ) ( RowBox SubscriptBox a 3 - SubscriptBox a 2 + 1 ) + RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( 1 ) ( SubscriptBox a 3 ) + RowBox UnderoverscriptBox RowBox k = 4 RowBox q + 1 RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( 1 ) ( RowBox SubscriptBox a k - SubscriptBox a 3 ) - RowBox UnderoverscriptBox RowBox k = 1 q RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( 1 ) ( RowBox SubscriptBox b k - SubscriptBox a 3 ) ) ) RowBox ( RowBox RowBox O ( FractionBox 1 z ) + 1 ) + RowBox UnderoverscriptBox RowBox k = 4 RowBox q + 1 ErrorBox RowBox FractionBox RowBox RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k RowBox q + 1 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 RowBox z ) RowBox RowBox SubscriptBox a 2 - SubscriptBox a 1 TagBox RowBox RowBox RowBox SubscriptBox a 3 - SubscriptBox a 2 TagBox Function Function RowBox RowBox SubscriptBox a k - SubscriptBox a 1 TagBox RowBox RowBox 4 k RowBox q + 1 Function RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 4 j RowBox q + 1 RowBox 4 k RowBox q + 1 RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) TraditionalForm [/itex]

 Rule Form

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

 2001-10-29