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HypergeometricU






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricU[a,b,z] > Specific values > Specialized values > For fixed z and with symbolical integers in parameters > For fixed z and a=n, b=1/2+-m





http://functions.wolfram.com/07.33.03.0057.01









  


  










Input Form





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










Standard Form





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MathML Form







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type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02