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variants of this functions
NorlundB




Mathematica Notation

Traditional Notation









Polynomials > NorlundB[n,α,z] > Series representations > Generalized power series > Expansions at generic point alpha==alpha0





http://functions.wolfram.com/05.17.06.0015.01









  


  








Input Form





NorlundB[n, \[Alpha], z] == Sum[Binomial[n, k] z^k Sum[(1/j!) Derivative[0, j][NorlundB][n - k, Subscript[\[Alpha], 0]] (\[Alpha] - Subscript[\[Alpha], 0])^j, {j, 0, n - k}], {k, 0, n}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["NorlundB", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", " ", "k"]], "]"]], SuperscriptBox["z", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[FractionBox["1", RowBox[List["j", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["0", ",", "j"]], "]"]], "[", "NorlundB", "]"]], "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", SubscriptBox["\[Alpha]", "0"]]], "]"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "-", SubscriptBox["\[Alpha]", "0"]]], ")"]], "j"]]]]]]]]]]]]]








MathML Form







<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> &#945; </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </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], Rule[Selectable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> j </mi> </msup> <semantics> <msubsup> <mi> B </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </msubsup> <annotation encoding='Mathematica'> TagBox[SubsuperscriptBox[TagBox[&quot;B&quot;, NorlundB], RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;k&quot;]], RowBox[List[&quot;(&quot;, &quot;\[Alpha]&quot;, &quot;)&quot;]]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#945; </mi> <mi> j </mi> </msup> </mrow> </mfrac> <msub> <mo> | </mo> <mrow> <mi> &#945; </mi> <mo> &#10869; </mo> <msub> <mi> &#945; </mi> <mn> 0 </mn> </msub> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> - </mo> <msub> <mi> &#945; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> NorlundB </ci> </apply> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> &#945; </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#63449; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <ms> n </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> &#8706; </ms> <ms> j </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> B </ms> <ci> NorlundB </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> &#945; </ms> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> $CellContext`e </ci> <apply> <ci> Zeta </ci> <ci> $CellContext`e </ci> </apply> </apply> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8706; </ms> <apply> <ci> SuperscriptBox </ci> <ms> &#945; </ms> <ms> j </ms> </apply> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> | </ms> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> &#10869; </ms> <apply> <ci> SubscriptBox </ci> <ms> &#945; </ms> <ms> 0 </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> &#945; </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> &#945; </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "\[Alpha]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "k"]]], FractionBox[RowBox[List[RowBox[List[SuperscriptBox["NorlundB", TagBox[RowBox[List["(", RowBox[List["0", ",", "j"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", SubscriptBox["\[Alpha]\[Alpha]", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "-", SubscriptBox["\[Alpha]\[Alpha]", "0"]]], ")"]], "j"]]], RowBox[List["j", "!"]]]]]]]]]]]]]


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





2007-05-02