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 HypergeometricU

 http://functions.wolfram.com/07.33.03.0065.01

 Input Form

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

 Standard Form

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 MathML Form

 U TagBox["U", HypergeometricU] ( n + 1 2 , m + 1 2 , z ) z ( ( - 1 ) m + n Γ ( 1 2 - n ) L n - m m - 1 2 ( - z ) + 1 ( m + 1 2 ) n - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["n", "-", "m"]]], Pochhammer] k = 0 n - m ( - 1 ) k k ! L - k - m + n k ( - z ) ( - erf ( z ) Γ ( k - m + 1 2 ) + - z j = 0 k - m - 1 z j + 1 2 ( k - m + 1 2 ) j - k + m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "m", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "k", "+", "m", "+", "1"]]], Pochhammer] - - z j = k - m - 1 z j + 1 2 ( k - m + 1 2 ) j - k + m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "m", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "k", "+", "m", "+", "1"]]], Pochhammer] ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n m m TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition HypergeometricU n 1 2 m 1 2 z z -1 m n Gamma 1 2 -1 n LaguerreL n -1 m m -1 1 2 -1 z 1 Pochhammer m 1 2 n -1 m -1 k 0 n -1 m -1 k k -1 LaguerreL -1 k -1 m n k -1 z -1 Erf z 1 2 Gamma k -1 m 1 2 -1 z j 0 k -1 m -1 z j 1 2 Pochhammer k -1 m 1 2 j -1 k m 1 -1 -1 -1 z j k -1 m -1 z j 1 2 Pochhammer k -1 m 1 2 j -1 k m 1 -1 n n m m [/itex]

 Rule Form

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 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02