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variants of this functions
EulerE




Mathematica Notation

Traditional Notation









Polynomials > EulerE[n,z] > Representations through more general functions > Through other functions > Involving zeta functions





http://functions.wolfram.com/05.13.26.0003.01









  


  








Input Form





EulerE[n, z] == 2^(n + 1) (Zeta[-n, z/2] - Zeta[-n, (z + 1)/2]) /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EulerE", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["n", "+", "1"]]], " ", RowBox[List["(", " ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox["z", "2"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<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> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, EulerE] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;n&quot;]], Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;z&quot;, &quot;2&quot;], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`a, $CellContext`b], Zeta[$CellContext`a, $CellContext`b]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;n&quot;]], Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`a, $CellContext`b], Zeta[$CellContext`a, $CellContext`b]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EulerE </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EulerE", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["n", "+", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox["z", "2"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29