BernoulliB[n, z] - BernoulliB[n] == n Sum[(((-1)^k Pochhammer[-z, k])/k) StirlingS2[n - 1, k - 1], {k, 1, n}]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", "z"]], "]"]], "-", RowBox[List["BernoulliB", "[", "n", "]"]]]], "\[Equal]", RowBox[List["n", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "z"]], ",", "k"]], "]"]]]], "k"], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["k", "-", "1"]]]], "]"]]]]]]]]]]]]

<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> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> </mrow> <mo> &#10869; </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;z&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <ci> BernoulliB </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> BernoulliB </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> StirlingS2 </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n_", ",", "z_"]], "]"]], "-", RowBox[List["BernoulliB", "[", "n_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["n", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "z"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["k", "-", "1"]]]], "]"]]]], "k"]]]]]]]]]

2001-10-29