NorlundB[n, z] == Sum[(((-1)^k Pochhammer[z, k])/Pochhammer[n + 1, k]) Subscript[b, n + k, k], {k, 1, n}] /; (Subscript[b, n, k] == 0 /; k > n/2 && k < 0) && Subscript[b, 0, 0] == 1 && Subscript[b, n, 1] == 1 && Subscript[b, n, k] == 1 && Subscript[b, n, k] == (n - 1) Subscript[b, n - 2, k - 1] + k Subscript[b, n - 1, k] && Element[k, Integers] && k >= 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> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, NorlundB] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#63449; </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> <mi> z </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;z&quot;, &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mfrac> <mo> &#8290; </mo> <msub> <mi> b </mi> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> , </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &gt; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#63449; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#63449; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> &#63449; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> &#63449; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <msub> <mi> b </mi> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> NorlundB </ci> <ci> n </ci> <ci> z </ci> </apply> <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> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <gt /> <ci> k </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02