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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0055.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", 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"], "+", "2"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "2"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "q"]], "/", "2"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "q"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["q", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "q"]], "-", RowBox[List["2", "k"]], "-", "1"]], ")"]], "!"]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["2", "k"]]], "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "-", RowBox[List["2", "k"]]]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]]]]], RowBox[List["BernoulliB", "[", RowBox[List["2", "k"]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "1"]]], SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "q"]], "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "q"]], "/", "2"]]]]]]]]], "/;", 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", ">", "0"]]]]]]]]

 MathML Form

 q + 1 F ~ q ( 1 , a 2 , , a q + 1 ; a 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"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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] π - q / 2 ( q - 1 ) ! k = 0 q 2 ( - 1 ) q - k - 1 ( 2 q - 2 k - 1 ) ! ( 2 2 k - 1 ) ( 2 π ) 2 k B TagBox["B", BernoulliB] 2 k ( q - 2 k ) ! ( 2 k ) ! + ( - 1 ) q - 1 2 2 q - 1 π - q / 2 /; a 2 a 3 a q + 1 1 2 q > 0 Condition HypergeometricPFQ Subscript a 2 2 Subscript a q 1 2 1 -1 q 2 -1 q -1 -1 k 0 q 2 -1 -1 q -1 k -1 2 q -1 2 k -1 2 2 k -1 2 2 k BernoulliB 2 k q -1 2 k 2 k -1 -1 q -1 2 2 q -1 -1 q 2 -1 Subscript a 2 Subscript a 3 Subscript a q 1 1 2 q 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", 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"], "+", "2"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "+", "2"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox["q", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["q", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "q"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["2", " ", "k"]]]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "q"]], ")"]], "!"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "q"]], "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox["q", "2"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "\[Equal]", SubscriptBox["a", "3"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", FractionBox["1", "2"]]], "&&", RowBox[List["q", ">", "0"]]]]]]]]]]

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

 2001-10-29