Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
BernoulliB






Mathematica Notation

Traditional Notation









Integer Functions > BernoulliB[n] > Identities > Functional identities





http://functions.wolfram.com/04.13.17.0005.01









  


  










Input Form





Sum[(BernoulliB[k] BernoulliB[n - k])/(k (n - k)), {k, 2, n - 2}] - Sum[(Binomial[n, k] BernoulliB[k] BernoulliB[n - k])/(k (n - k)), {k, 2, n - 2}] == 2 HarmonicNumber[n] (BernoulliB[n]/n) /; Element[n, Integers] && n >= 3










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n", "-", "k"]], "]"]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], RowBox[List["BernoulliB", "[", "k", "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n", "-", "k"]], "]"]]]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]]]]]], "\[Equal]", RowBox[List["2", " ", RowBox[List["HarmonicNumber", "[", "n", "]"]], FractionBox[RowBox[List[" ", RowBox[List["BernoulliB", "[", "n", "]"]]]], "n"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "3"]]]]]]]]










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> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mfrac> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> </mrow> <mi> n </mi> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mn> 3 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> BernoulliB </ci> <ci> k </ci> </apply> <apply> <ci> BernoulliB </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <ci> BernoulliB </ci> <ci> k </ci> </apply> <apply> <ci> BernoulliB </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> HarmonicNumber </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <ci> BernoulliB </ci> <ci> n </ci> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "2"]], RowBox[List["n_", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", "k_", "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n_", "-", "k_"]], "]"]]]], RowBox[List["k_", " ", RowBox[List["(", RowBox[List["n_", "-", "k_"]], ")"]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "2"]], RowBox[List["n_", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], " ", RowBox[List["BernoulliB", "[", "k_", "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n_", "-", "k_"]], "]"]]]], RowBox[List["k_", " ", RowBox[List["(", RowBox[List["n_", "-", "k_"]], ")"]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["HarmonicNumber", "[", "n", "]"]], " ", RowBox[List["BernoulliB", "[", "n", "]"]]]], "n"], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "3"]]]]]]]]]]










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





2002-12-18