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http://functions.wolfram.com/04.13.23.0003.01
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Sum[((2 - 2^(n - k + 2)) BernoulliB[n - k + 1] z^k)/(k! (n - k + 1)!),
{k, 0, n}] == (1/n!) EulerE[n, z]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "-", SuperscriptBox["2", RowBox[List["n", "-", "k", "+", "2"]]]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n", "-", "k", "+", "1"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "+", "1"]], ")"]], "!"]]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], RowBox[List["EulerE", "[", RowBox[List["n", ",", "z"]], "]"]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", EulerE] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BernoulliB </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> EulerE </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "n_"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "-", SuperscriptBox["2", RowBox[List["n_", "-", "k_", "+", "2"]]]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n_", "-", "k_", "+", "1"]], "]"]], " ", SuperscriptBox["z_", "k_"]]], RowBox[List[RowBox[List["k_", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "-", "k_", "+", "1"]], ")"]], "!"]]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["EulerE", "[", RowBox[List["n", ",", "z"]], "]"]], RowBox[List["n", "!"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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