Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Polynomials > LaguerreL[n,lambda,z] > Series representations > Generalized power series > Expansions at generic point lambda==lambda0 > For the function itself





http://functions.wolfram.com/05.08.06.0011.01









  


  








Input Form





LaguerreL[n, \[Lambda], z] \[Proportional] ((-1)^n z^n)/n! + (1/n!) Sum[(((-1)^(j + n - s) z^s)/s!) Pochhammer[-n, s] StirlingS1[n - s, j] (1 + s + Subscript[\[Lambda], 0])^j (1 + (j/(1 + s + Subscript[\[Lambda], 0])) (\[Lambda] - Subscript[\[Lambda], 0]) + (((-1 + j) j)/(2 (1 + s + Subscript[\[Lambda], 0])^2)) (\[Lambda] - Subscript[\[Lambda], 0])^2 + \[Ellipsis]), {s, 0, n}, {j, 1, n - s}] /; (\[Lambda] -> Subscript[\[Lambda], 0])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", "n"]]], RowBox[List["n", "!"]]], "+", RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "s"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n", "-", "s"]]], " ", SuperscriptBox["z", "s"], " "]], RowBox[List["s", "!"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "s"]], "]"]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["n", "-", "s"]], ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "s", "+", SubscriptBox["\[Lambda]", "0"]]], ")"]], "j"], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List["j", " "]], RowBox[List["(", RowBox[List["1", "+", "s", "+", SubscriptBox["\[Lambda]", "0"]]], ")"]]], " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", SubscriptBox["\[Lambda]", "0"]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "j"]], ")"]], " ", "j"]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "s", "+", SubscriptBox["\[Lambda]", "0"]]], ")"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "-", SubscriptBox["\[Lambda]", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]]]]]]]], " ", "/;", RowBox[List["(", RowBox[List["\[Lambda]", "\[Rule]", SubscriptBox["\[Lambda]", "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> <mi> L </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> s </mi> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> s </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;n&quot;]], &quot;)&quot;]], &quot;s&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> s </mi> </mrow> <mrow> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <msub> <mi> &#955; </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> j </mi> <mtext> </mtext> </mrow> <mrow> <mi> s </mi> <mo> + </mo> <msub> <mi> &#955; </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <msub> <mi> &#955; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <msub> <mi> &#955; </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <msub> <mi> &#955; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> &#955; </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> &#955; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> s </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> s </ci> </apply> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> s </ci> <apply> <ci> Subscript </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <ci> j </ci> <apply> <power /> <apply> <plus /> <ci> s </ci> <apply> <ci> Subscript </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> s </ci> <apply> <ci> Subscript </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> &#955; </ci> <apply> <ci> Subscript </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", "n"]]], RowBox[List["n", "!"]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "s"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n", "-", "s"]]], " ", SuperscriptBox["z", "s"]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "s"]], "]"]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["n", "-", "s"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "s", "+", SubscriptBox["\[Lambda]\[Lambda]", "0"]]], ")"]], "j"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["j", " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", SubscriptBox["\[Lambda]\[Lambda]", "0"]]], ")"]]]], RowBox[List["1", "+", "s", "+", SubscriptBox["\[Lambda]\[Lambda]", "0"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "j"]], ")"]], " ", "j"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "-", SubscriptBox["\[Lambda]\[Lambda]", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "s", "+", SubscriptBox["\[Lambda]\[Lambda]", "0"]]], ")"]], "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["s", "!"]]]]]]], RowBox[List["n", "!"]]]]], "/;", RowBox[List["(", RowBox[List["\[Lambda]", "\[Rule]", SubscriptBox["\[Lambda]\[Lambda]", "0"]]], ")"]]]]]]]]


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





2007-05-02