ParabolicCylinderD[2 n + 1, z] == ((-1)^n 2^n n! z LaguerreL[n, 1/2, z^2/2])/ E^(z^2/4) /; Element[n, Integers] && n >= 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msubsup> <mo> ( </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <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> <ci> ParabolicCylinderD </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <ci> z </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02