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 WhittakerM

 http://functions.wolfram.com/07.44.03.0060.01

 Input Form

 WhittakerM[-((2 m - 1)/4) - n, -((2 m + 1)/4), z] == (((-1)^m Sqrt[Pi] z^((1/4) (1 - 2 m)) (m + 1)!)/ (E^(z/2) (Gamma[1/2 + m] (m + n)!))) (((E^z Sqrt[Pi] 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}]) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0

 Standard Form

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 MathML Form

 M TagBox["M", WhittakerM] - 2 m - 1 4 - n , - 2 m + 1 4 ( z ) ( - 1 ) m - z 2 π z 1 - 2 m 4 ( m + 1 ) ! Γ ( m + 1 2 ) ( m + n ) ! ( z π erf ( z ) z k = 0 m + 1 ( k + m + n ) ! k ! L m - k + 1 k - m - 1 2 ( z ) L k + m + n - k - 1 2 ( - z ) + k = 0 m + 1 ( k + m + n ) ! k ! L m - k + 1 k - m - 1 2 ( z ) p = 1 k + m + n 1 p L k + m + n - p p - k - 1 2 ( - z ) L p - 1 1 2 - p ( z ) ) /; n m Condition WhittakerM -1 2 m -1 4 -1 -1 n -1 2 m 1 4 -1 z -1 m -1 z 2 -1 1 2 z 1 -1 2 m 4 -1 m 1 Gamma m 1 2 m n -1 z 1 2 Erf z 1 2 z 1 2 -1 k 0 m 1 k m n k -1 LaguerreL m -1 k 1 k -1 m -1 1 2 z LaguerreL k m n -1 k -1 1 2 -1 z k 0 m 1 k m n k -1 LaguerreL m -1 k 1 k -1 m -1 1 2 z p 1 k m n 1 p -1 LaguerreL k m n -1 p p -1 k -1 1 2 -1 z LaguerreL p -1 1 2 -1 p z n m [/itex]

 Rule Form

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

 2007-05-02