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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.06.0019.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (Pi^q/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) Sum[(((Gamma[Subscript[a, k]] Product[If[j != k, Csc[Pi (Subscript[a, j] - Subscript[a, k])], 1], {j, 1, q + 1}])/ Product[Gamma[Subscript[b, j] - Subscript[a, k]], {j, 1, q}]) HypergeometricPFQRegularized[{Subscript[a, k], Subscript[a, k] - Subscript[b, 1] + 1, \[Ellipsis], Subscript[a, k] - Subscript[b, q] + 1}, {Subscript[a, k] - Subscript[a, 1] + 1, \[Ellipsis], Subscript[a, k] - Subscript[a, k - 1] + 1, Subscript[a, k] - Subscript[a, k + 1] + 1, \[Ellipsis], Subscript[a, k] - Subscript[a, q + 1] + 1}, 1/z])/ (-z)^Subscript[a, k], {k, 1, q + 1}] /; !IntervalMemberQ[Interval[{0, 1}], z] && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= q + 1 && 1 <= k <= q + 1, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

 Standard Form

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 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] π q k = 1 q + 1 Γ ( a k ) k = 1 q + 1 Γ ( a k ) j = 1 j k q + 1 csc ( π ( a j - a k ) ) j = 1 q Γ ( b j - a k ) ( - z ) - a k q + 1 F ~ q ( a k , a k - b 1 + 1 , , a k - b q + 1 ; 1 - a 1 + a k , , 1 - a k - 1 + a k , 1 - a k + 1 + a k , , 1 - a q + 1 + a k ; 1 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", "k"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "q"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; z ( 0 , 1 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j q + 1 1 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 SuperscriptBox π q RowBox UnderoverscriptBox RowBox k = 1 RowBox q + 1 RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox RowBox k = 1 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 csc ( RowBox π RowBox ( 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 TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox RowBox SubscriptBox a k - SubscriptBox b 1 + 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox SubscriptBox a k - SubscriptBox b q + 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox a 1 + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox k - 1 + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox k + 1 + SubscriptBox a k HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox q + 1 + SubscriptBox a k HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox FractionBox 1 z HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ /; RowBox RowBox z RowBox ( RowBox 0 , 1 ) RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 1 j RowBox q + 1 RowBox 1 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