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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LaguerreL[nu,lambda,z] > Summation > Infinite summation





http://functions.wolfram.com/07.03.23.0003.01









  


  








Input Form





Sum[(LaguerreL[n, k + \[Lambda], z] w^k)/k!, {k, 0, Infinity}] == E^w LaguerreL[n, \[Lambda], z - w] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List["k", "+", "\[Lambda]"]], ",", "z"]], "]"]], " ", SuperscriptBox["w", "k"]]], RowBox[List["k", "!"]]]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["z", "-", "w"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#955; </mi> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> k </mi> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> &#8519; </mi> <mi> w </mi> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> LaguerreL </ci> <ci> n </ci> <apply> <plus /> <ci> k </ci> <ci> &#955; </ci> </apply> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> &#955; </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", RowBox[List["k_", "+", "\[Lambda]_"]], ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "k_"]]], RowBox[List["k_", "!"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["z", "-", "w"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29