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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.03.0057.01

 Input Form

 HypergeometricPFQ[{1, Subscript[a, 2], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[a, 2] + 1, \[Ellipsis], Subscript[a, n + 1] + 1, Subscript[a, n + 2] + 2, \[Ellipsis], Subscript[a, q + 1] + 2}, 1] == (-1)^q 3^(q - n) 2^(n - 2 q) (Sum[Binomial[q + k - 1, k] (2^(q - n - k) - 1) Zeta[q - n - k], {k, 0, q - n - 2}] + Sum[(-1)^(q - k) Binomial[q - n + k - 1, k] (2^(q - k) - 1) Zeta[q - k], {k, 0, q - 2}] - Sum[Binomial[q + k - 1, k] 2^(q - n - k), {k, 0, q - n - 1}]) /; Subscript[a, 2] == Subscript[a, 3] == \[Ellipsis] == Subscript[a, q + 1] == 1/2 && q > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "1"]], ",", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "2"]]], "+", "2"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "2"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], SuperscriptBox["3", RowBox[List["q", "-", "n"]]], SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", "q"]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "n", "-", "2"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "+", "k", "-", "1"]], ",", "k"]], "]"]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["q", "-", "n", "-", "k"]]], "-", "1"]], ")"]], RowBox[List["Zeta", "[", RowBox[List["q", "-", "n", "-", "k"]], "]"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "k"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "-", "n", "+", "k", "-", "1"]], ",", "k"]], "]"]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["q", "-", "k"]]], "-", "1"]], ")"]], RowBox[List["Zeta", "[", RowBox[List["q", "-", "k"]], "]"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "+", "k", "-", "1"]], ",", "k"]], "]"]], SuperscriptBox["2", RowBox[List["q", "-", "n", "-", "k"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "\[Equal]", SubscriptBox["a", "3"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", FractionBox["1", "2"]]], "\[And]", RowBox[List["q", ">", "1"]]]]]]]]

 MathML Form

 q + 1 F q ( 1 , a 2 , , a q + 1 ; a 2 + 1 , , a n + 1 + 1 , a n + 2 + 2 , , a q + 1 + 2 ; 1 ) 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["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "1"]], ",", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "2"]]], "+", "2"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( - 1 ) q 3 q - n 2 n - 2 q ( k = 0 q - n - 2 ( k + q - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "q", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 q - n - k - 1 ) ζ ( q - n - k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["q", "-", "n", "-", "k"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] + k = 0 q - 2 ( - 1 ) q - k ( k - n + q - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", "n", "+", "q", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 q - k - 1 ) ζ ( q - k ) - k = 0 q - n - 1 ( k + q - 1 k ) 2 q - n - k TagBox[RowBox[List[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["q", "-", "k"]], Rule[Editable, True]], ")"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "n", "-", "1"]]], RowBox[List[TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "q", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]], " ", SuperscriptBox["2", RowBox[List["q", "-", "n", "-", "k"]]]]]]]]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] ) /; a 2 a 3 a q + 1 1 2 q > 1 FormBox RowBox RowBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox F FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox 1 , SubscriptBox a 2 , , SubscriptBox a RowBox q + 1 InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox RowBox SubscriptBox a 2 + 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox ErrorBox RowBox RowBox SubscriptBox a RowBox n + 1 + 1 , RowBox SubscriptBox a RowBox n + 2 + 2 , , RowBox SubscriptBox a RowBox q + 1 + 2 HypergeometricPFQ Rule Editable Function Null HoldComplete SlotSequence 1 HoldAllComplete HypergeometricPFQ Rule Editable ; TagBox 1 HypergeometricPFQ Rule Editable ) Function Null HoldComplete HypergeometricPFQ Slot 1 Slot 2 Slot 3 HoldAllComplete RowBox SuperscriptBox RowBox ( RowBox - 1 ) q SuperscriptBox 3 RowBox q - n SuperscriptBox 2 RowBox n - RowBox 2 q RowBox ( RowBox RowBox UnderoverscriptBox RowBox k = 0 RowBox q - n - 2 RowBox TagBox RowBox ( GridBox TagBox RowBox k + q - 1 Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox ( RowBox SuperscriptBox 2 RowBox q - n - k - 1 ) TagBox RowBox ζ ( TagBox RowBox q - n - k Rule Editable ) InterpretTemplate Function \$CellContext`e Zeta \$CellContext`e + RowBox UnderoverscriptBox RowBox k = 0 RowBox q - 2 RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox q - k TagBox RowBox ( GridBox TagBox RowBox k - n + q - 1 Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox ( RowBox SuperscriptBox 2 RowBox q - k - 1 ) TagBox RowBox RowBox ζ ( TagBox RowBox q - k Rule Editable ) - RowBox UnderoverscriptBox RowBox k = 0 RowBox q - n - 1 RowBox TagBox RowBox ( GridBox TagBox RowBox k + q - 1 Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable SuperscriptBox 2 RowBox q - n - k InterpretTemplate Function \$CellContext`e Zeta \$CellContext`e ) /; RowBox RowBox SubscriptBox a 2 SubscriptBox a 3 SubscriptBox a RowBox q + 1 FractionBox 1 2 RowBox q > 1 TraditionalForm [/itex]

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

 2001-10-29