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http://functions.wolfram.com/05.14.16.0011.01
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BernoulliB[n, m z] == m^(n - 1) Sum[BernoulliB[n, k/m + z], {k, 0, m - 1}] /;
Element[m, Integers] && m > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List["m", " ", "z"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List[FractionBox["k", "m"], "+", "z"]]]], "]"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["m", " ", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "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> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> m </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mi> k </mi> <mi> m </mi> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BernoulliB </ci> <ci> n </ci> <apply> <times /> <ci> m </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> m </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <ci> BernoulliB </ci> <ci> n </ci> <apply> <plus /> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BernoulliB", "[", RowBox[List["n_", ",", RowBox[List["m_", " ", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List[FractionBox["k", "m"], "+", "z"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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