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variants of this functions
LaguerreL




Mathematica Notation

Traditional Notation









Polynomials > LaguerreL[n,lambda,z] > Summation > Infinite summation





http://functions.wolfram.com/05.08.23.0011.01









  


  








Input Form





Sum[(t (1 - t)^\[Mu] E^((\[Alpha] z t)/(1 - t)))^k LaguerreL[k, \[Lambda] + \[Mu] k, z (1 + \[Alpha] k)], {k, 0, Infinity}] == ((1 - t)^(1 - \[Lambda]) E^((z t)/(t - 1)))/ (1 - t (\[Mu] + 2 - \[Alpha] z) + t^2 (\[Mu] + 1))








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mi> k </mi> <mrow> <mi> &#955; </mi> <mo> + </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#955; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mrow> <mi> t </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <times /> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> &#945; </ci> <ci> z </ci> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> LaguerreL </ci> <ci> k </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <ci> k </ci> <ci> &#956; </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> &#945; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <ci> t </ci> <apply> <power /> <apply> <plus /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> &#945; </ci> </apply> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <ci> t </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18