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variants of this functions
BernoulliB






Mathematica Notation

Traditional Notation









Polynomials > BernoulliB[n,z] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/05.14.17.0006.01









  


  










Input Form





BernoulliB[n, m] == BernoulliB[n, 0] + n Sum[k^(n - 1), {k, 1, m - 1}] /; Element[m, Integers] && m >= 0 && Element[n, Integers] && n > 1










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", "0"]], "]"]], "+", RowBox[List["n", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], SuperscriptBox["k", RowBox[List["n", "-", "1"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "1"]]]]]]]]










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> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <msup> <mi> k </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BernoulliB </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> BernoulliB </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <ci> n </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <power /> <ci> k </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BernoulliB", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", "0"]], "]"]], "+", RowBox[List["n", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], SuperscriptBox["k", RowBox[List["n", "-", "1"]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "1"]]]]]]]]]]










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





2001-10-29