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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.06.0016.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, q}]/Product[Gamma[Subscript[a, k]], {k, 1, q + 1}]) 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 + (Subscript[a, k] Product[Subscript[a, k] - Subscript[b, j] + 1, {j, 1, q}])/Product[If[j != k, Subscript[a, k] - Subscript[a, j] + 1, 1] z, {j, 1, q + 1}] + (Subscript[a, k] (Subscript[a, k] + 1) Product[(Subscript[a, k] - Subscript[b, j] + 1) (Subscript[a, k] - Subscript[b, j] + 2), {j, 1, q}])/ (2 Product[If[j != k, (Subscript[a, k] - Subscript[a, j] + 1) (Subscript[a, k] - Subscript[a, j] + 2), 1] z^2, {j, 1, q + 1}]) + \[Ellipsis]))/(-z)^Subscript[a, k], {k, 1, q + 1}] /; Abs[z] > 1 && 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]], " ", ",", "1"]], "]"]]]]]], 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", "+", FractionBox[RowBox[List[SubscriptBox["a", "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "j"], "+", "1"]], ")"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "j"], "+", "1"]], ",", "1"]], "]"]], "z"]]]]], "+", FractionBox[RowBox[List[SubscriptBox["a", "k"], RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "+", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "j"], "+", "1"]], ")"]], RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "j"], "+", "2"]], ")"]]]], ")"]], " "]]]], RowBox[List["2", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["q", "+", "1"]]], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[NotEqual]", "k"]], ",", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "j"], "+", "1"]], ")"]], RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "j"], "+", "2"]], ")"]]]], ",", "1"]], "]"]], SuperscriptBox["z", "2"]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]], "\[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]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]], RowBox[List["(", "\[InvisibleSpace]", 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["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] k = 1 q Γ ( b k ) k = 1 q + 1 Γ ( a k ) k = 1 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 + a k j = 1 q ( a k - b j + 1 ) j = 1 j k q + 1 ( a k - a j + 1 ) z + a k ( a k + 1 ) j = 1 q ( a k - b j + 1 ) ( a k - b j + 2 ) 2 j = 1 j k q + 1 ( ( a k - a j + 1 ) ( a k - a j + 2 ) ) z 2 + ) /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 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 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 RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( SubscriptBox b k ) 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 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 + FractionBox RowBox SubscriptBox a k RowBox UnderoverscriptBox RowBox j = 1 q RowBox ( RowBox SubscriptBox a k - SubscriptBox b j + 1 ) RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k RowBox q + 1 RowBox RowBox ( RowBox SubscriptBox a k - SubscriptBox a j + 1 ) z + FractionBox RowBox RowBox SubscriptBox a k ( RowBox SubscriptBox a k + 1 ) RowBox UnderoverscriptBox RowBox j = 1 q RowBox RowBox ( RowBox SubscriptBox a k - SubscriptBox b j + 1 ) RowBox ( RowBox SubscriptBox a k - SubscriptBox b j + 2 ) RowBox 2 RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k RowBox q + 1 RowBox RowBox ( RowBox RowBox ( RowBox SubscriptBox a k - SubscriptBox a j + 1 ) RowBox ( RowBox SubscriptBox a k - SubscriptBox a j + 2 ) ) SuperscriptBox z 2 + ) /; RowBox RowBox RowBox z > 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]

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

 2001-10-29