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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 z==infinity





http://functions.wolfram.com/05.17.06.0010.01









  


  








Input Form





NorlundB[n, \[Alpha], z] == z^n (1 - (\[Alpha] n)/(2 z) + (\[Alpha] (3 \[Alpha] - 1) (n - 1) n)/(24 z^2) + \[Ellipsis]) /; (Abs[z] -> Infinity) && n > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["NorlundB", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", "n"], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[Alpha]", " ", "n"]], RowBox[List["2", " ", "z", " "]]], "+", FractionBox[RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Alpha]"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n"]], RowBox[List["24", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]








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> <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> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> NorlundB </ci> <ci> n </ci> <ci> &#945; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#945; </ci> <ci> n </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#945; </ci> <ci> n </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </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[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[Alpha]", " ", "n"]], RowBox[List["2", " ", "z"]]], "+", FractionBox[RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Alpha]"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n"]], RowBox[List["24", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]


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





2007-05-02