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variants of this functions
NorlundB






Mathematica Notation

Traditional Notation









Polynomials > NorlundB[n,α,z] > Differentiation > Symbolic differentiation > With respect to alpha





http://functions.wolfram.com/05.17.20.0004.01









  


  










Input Form





D[NorlundB[n, \[Alpha], z], {\[Alpha], m}] == n! Sum[(Subscript[c, n - k, m]/(k! (n - k)!)) z^k, {k, 0, n}] /; Subscript[c, k, m] == Sum[(-1)^j Binomial[k, j] Sum[Binomial[m, r] (m - r)! \[Alpha]^(1 - r) (\[Alpha] + j)^(-1 - m + r) Sum[(-1)^(i + k + m - r) StirlingS1[k + 1, i + 1] Pochhammer[i - r + 2, r] \[Alpha]^i Subscript[p, j, k], {i, 0, k}], {r, 0, m}], {j, 0, k}] && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) Subscript[a, m] Subscript[p, j, k - m], {m, 1, k}] && Subscript[a, k] == 1/(k + 1)! && Element[k, Integers] && k >= 0










Standard Form





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MathML Form







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</mo> </mrow> </mfrac> </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> <partialdiff /> <bvar> <ci> &#945; </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <ci> NorlundB </ci> <ci> n </ci> <ci> &#945; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> <ci> m </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Binomial </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> m </ci> <ci> r </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> j </ci> <ci> &#945; </ci> </apply> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> i </ci> <ci> k </ci> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <cn type='integer'> 2 </cn> </apply> <ci> r </ci> </apply> <apply> <power /> <ci> &#945; </ci> <ci> i </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> j </ci> <ci> m </ci> </apply> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02