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http://functions.wolfram.com/07.32.03.0070.01
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HypergeometricPFQRegularized[{1, Subscript[a, 2], \[Ellipsis],
Subscript[a, n + 1], Subscript[a, n + 2], \[Ellipsis],
Subscript[a, q + 1]}, {Subscript[a, 2] + 2, \[Ellipsis],
Subscript[a, n + 1] + 2, Subscript[a, n + 2] + 1, \[Ellipsis],
Subscript[a, q + 1] + 1}, -1] ==
((-1)^(n - 1) 2^(-n + q) (Sum[Binomial[n + k - 1, k] 2^(k - q)
(Zeta[q - k, 1/4] - Zeta[q - k, 3/4]), {k, 0, q - 2}] -
Sum[(-1)^(n - k) Binomial[q + k - 1, k] 2^(k - n)
(Zeta[n - k, 1/4] - Zeta[n - k, 3/4]), {k, 0, n - 2}] -
Sum[Binomial[n + k - 1, k] 2^(q - k), {k, 0, q - 1}] +
Pi Binomial[q + n - 2, n - 1]))/Pi^(q/2) /;
Subscript[a, 2] == Subscript[a, 3] == \[Ellipsis] == Subscript[a, n + 1] ==
1/2 && Subscript[a, n + 2] == Subscript[a, n + 3] == \[Ellipsis] ==
Subscript[a, q + 1] == 3/2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", SubscriptBox["a", RowBox[List["n", "+", "2"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "+", "2"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "2"]], ",", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "2"]]], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "q"]], "/", "2"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "2"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "1"]], ",", "k"]], "]"]], SuperscriptBox["2", RowBox[List["k", "-", "q"]]], RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["q", "-", "k"]], ",", FractionBox["1", "4"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["q", "-", "k"]], ",", FractionBox["3", "4"]]], "]"]]]], ")"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "+", "k", "-", "1"]], ",", "k"]], "]"]], SuperscriptBox["2", RowBox[List["k", "-", "n"]]], RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", FractionBox["1", "4"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", FractionBox["3", "4"]]], "]"]]]], ")"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "1"]], ",", "k"]], "]"]], SuperscriptBox["2", RowBox[List["q", "-", "k"]]]]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "+", "n", "-", "2"]], ",", RowBox[List["n", "-", "1"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "\[Equal]", SubscriptBox["a", "3"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox["1", "2"]]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "2"]]], "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "3"]]], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", FractionBox["3", "2"]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", SubscriptBox["a", RowBox[List["n", "+", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "2"]], ",", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "2"]]], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]]]], 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] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> q </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "+", "q", "-", "2"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> q </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> 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]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["n", "-", "k"]], Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["n", "-", "k"]], Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> k </mi> <mo> - 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Date Added to functions.wolfram.com (modification date)
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