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http://functions.wolfram.com/05.14.03.0004.01
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BernoulliB[n, 1/4] == (-2^(-n)) (1 - 2^(1 - n)) BernoulliB[n] -
(n EulerE[n - 1])/4^n /; Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", FractionBox["1", "4"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "n"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["1", "-", "n"]]]]], ")"]], RowBox[List["BernoulliB", "[", "n", "]"]]]], "-", RowBox[List["n", " ", SuperscriptBox["4", RowBox[List["-", "n"]]], RowBox[List["EulerE", "[", RowBox[List["n", "-", "1"]], "]"]]]]]]]], " ", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]
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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> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", EulerE] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <msup> <mn> 4 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BernoulliB </ci> <ci> n </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> EulerE </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> BernoulliB </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BernoulliB", "[", RowBox[List["n_", ",", FractionBox["1", "4"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "n"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["1", "-", "n"]]]]], ")"]], " ", RowBox[List["BernoulliB", "[", "n", "]"]]]], "-", RowBox[List["n", " ", SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List["EulerE", "[", RowBox[List["n", "-", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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