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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0069.01

 Input Form

 HypergeometricPFQRegularized[{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 - 1) (Sum[Binomial[q + k - 1, k] 2^(k - q + n) (Zeta[q - n - k, 1/4] - Zeta[q - n - k, 3/4]), {k, 0, q - n - 2}] - Sum[(-1)^(q - k) Binomial[q - n + k - 1, k] 2^(k - q) (Zeta[q - k, 1/4] - Zeta[q - k, 3/4]), {k, 0, q - 2}] - Sum[Binomial[q + k - 1, k] 2^(q - n - k), {k, 0, q - n - 1}] + Pi Binomial[2 q - n - 2, q - 1]))/Pi^(q/2) /; Subscript[a, 2] == Subscript[a, 3] == \[Ellipsis] == Subscript[a, q + 1] == 1/2 && q > 1

 Standard Form

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 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[OverscriptBox["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[RowBox[List["-", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( - 1 ) q - 1 π - q / 2 ( π ( 2 q - n - 2 q - 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "q"]], "-", "n", "-", "2"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["q", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] - k = 0 q - n - 1 ( 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 + 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 k + n - q ( ζ ( q - n - k , 1 4 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["q", "-", "n", "-", "k"]], Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[\$CellContext`e1, \$CellContext`e2], Zeta[\$CellContext`e1, \$CellContext`e2]]]] - ζ ( q - n - k , 3 4 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["q", "-", "n", "-", "k"]], Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[\$CellContext`e1, \$CellContext`e2], Zeta[\$CellContext`e1, \$CellContext`e2]]]] ) - 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 k - q ( ζ ( q - k , 1 4 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["q", "-", "k"]], Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[\$CellContext`e1, \$CellContext`e2], Zeta[\$CellContext`e1, \$CellContext`e2]]]] - ζ ( q - k , 3 4 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["q", "-", "k"]], Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[\$CellContext`e1, \$CellContext`e2], Zeta[\$CellContext`e1, \$CellContext`e2]]]] ) ) /; a 2 a 3 a q + 1 1 2 q > 1 FormBox RowBox RowBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox OverscriptBox 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 RowBox - 1 HypergeometricPFQ Rule Editable ) Function Null HoldComplete HypergeometricPFQ Slot 1 Slot 2 Slot 3 HoldAllComplete RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox q - 1 SuperscriptBox π RowBox RowBox - q / 2 RowBox ( RowBox RowBox π TagBox RowBox ( GridBox TagBox RowBox RowBox 2 q - n - 2 Rule Editable TagBox RowBox q - 1 Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 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 + 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 SuperscriptBox 2 RowBox k + n - q RowBox ( RowBox TagBox RowBox ζ ( RowBox TagBox RowBox q - n - k Rule Editable , TagBox FractionBox 1 4 Rule Editable ) InterpretTemplate \$CellContext`e1 \$CellContext`e2 Zeta \$CellContext`e1 \$CellContext`e2 - TagBox RowBox ζ ( RowBox TagBox RowBox q - n - k Rule Editable , TagBox FractionBox 3 4 Rule Editable ) InterpretTemplate \$CellContext`e1 \$CellContext`e2 Zeta \$CellContext`e1 \$CellContext`e2 ) - 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 SuperscriptBox 2 RowBox k - q RowBox ( RowBox TagBox RowBox ζ ( RowBox TagBox RowBox q - k Rule Editable , TagBox FractionBox 1 4 Rule Editable ) InterpretTemplate \$CellContext`e1 \$CellContext`e2 Zeta \$CellContext`e1 \$CellContext`e2 - TagBox RowBox ζ ( RowBox TagBox RowBox q - k Rule Editable , TagBox FractionBox 3 4 Rule Editable ) InterpretTemplate \$CellContext`e1 \$CellContext`e2 Zeta \$CellContext`e1 \$CellContext`e2 ) ) /; RowBox RowBox SubscriptBox a 2 SubscriptBox a 3 SubscriptBox a RowBox q + 1 FractionBox 1 2 RowBox q > 1 TraditionalForm [/itex]

 Rule Form

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

 2001-10-29