WhittakerM[-((2 m - 1)/4) + n, -((2 m - 1)/4), z] == (((-1)^m z^((3 - 2 m)/4) m!)/(E^(z/2) (2 n! Pochhammer[-(1/2), m]))) (((Sqrt[Pi] Erfi[Sqrt[z]])/Sqrt[z]) Sum[((k + n)!/k!) LaguerreL[-k + m, 1/2 + k - m, -z] LaguerreL[k + n, -(1/2) - k, z], {k, 0, m}] + E^z Sum[((k + n)!/k!) LaguerreL[-k + m, 1/2 + k - m, -z] Sum[(1/p) LaguerreL[k + n - p, -(1/2) - k + p, z] LaguerreL[-1 + p, 1/2 - p, -z], {p, 1, k + n}], {k, 0, m}]) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 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> M </mi> <annotation encoding='Mathematica'> TagBox[&quot;M&quot;, WhittakerM] </annotation> </semantics> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mtext> </mtext> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </mrow> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> 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</mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mi> z 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<mo> - </mo> <mi> p </mi> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> p </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> WhittakerM </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn 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type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> 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Brychkov Yu.A. (2006)

2007-05-02