HypergeometricU[n + 1/2, m, z] == (((-1)^(-1 + m) (-1 + m)!)/(Sqrt[Pi] Pochhammer[1/2, n]^2)) z^(1 - m) E^(z/2) Sum[((k + n)!/k!) LaguerreL[-1 - k + m, 3/2 + k - m, z] Sum[(((-1)^p z^p)/(2^(2 p) p!)) LaguerreL[k + n - p, -(1/2) - k + p, -(z/2)] Sum[Binomial[p, j] BesselK[-2 j + p, z/2], {j, 0, p}], {p, 0, k + n}], {k, 0, -1 + m}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

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</mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> p </mi> </msup> </mrow> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> </mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> p </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;p&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mrow> <mi> p </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mrow> </msub> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </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> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricU </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> m </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -2 </cn> <ci> p </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <ci> BesselK </ci> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; 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Brychkov Yu.A. (2006)

2007-05-02