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 HypergeometricU

 http://functions.wolfram.com/07.33.03.0063.01

 Input Form

 HypergeometricU[n + 1/2, 3/2, z] == (1/Pochhammer[1/2, n]) ((Sqrt[Pi]/(2 z)) E^z (LaguerreL[-1 + n, -(1/2), -z] + 2 n LaguerreL[n, -(3/2), -z]) Erf[Sqrt[z]] + (1/(2 Sqrt[z])) Sum[(1/(p + 1)) LaguerreL[-2 + n - p, 1/2 + p, -z] LaguerreL[p, -(1/2) - p, z], {p, 0, -2 + n}] + (n/Sqrt[z]) Sum[(1/(p + 1)) LaguerreL[n - p - 1, -(1/2) + p, -z] LaguerreL[p, -(1/2) - p, z], {p, 0, n - 1}]) - (-1)^n E^z Gamma[1/2 - n] LaguerreL[-1 + n, 1/2, -z] /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", FractionBox["3", "2"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SqrtBox["\[Pi]"], RowBox[List["2", "z"]]], SuperscriptBox["\[ExponentialE]", "z"], RowBox[List["(", RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", "z"]]]], "]"]], "+", RowBox[List["2", "n", " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]], RowBox[List["Erf", "[", SqrtBox["z"], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "+", "n"]]], RowBox[List[FractionBox["1", RowBox[List["p", "+", "1"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "n", "-", "p"]], ",", RowBox[List[FractionBox["1", "2"], "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "p"]], ",", "z"]], "]"]]]]]]]], "+", RowBox[List[FractionBox["n", SqrtBox["z"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["p", "+", "1"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", "p", "-", "1"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["p", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "p"]], ",", "z"]], "]"]]]]]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", FractionBox["1", "2"], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 U TagBox["U", HypergeometricU] ( n + 1 2 , 3 2 , z ) 1 ( 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] ( π 2 z z erf ( z ) ( L n - 1 - 1 2 ( - z ) + 2 n L n - 3 2 ( - z ) ) + n z p = 0 n - 1 1 p + 1 L n - p - 1 p - 1 2 ( - z ) L p - p - 1 2 ( z ) + 1 2 z p = 0 n - 2 1 p + 1 L n - p - 2 p + 1 2 ( - z ) L p - p - 1 2 ( z ) ) - ( - 1 ) n z Γ ( 1 2 - n ) L n - 1 1 2 ( - z ) /; n + Condition HypergeometricU n 1 2 3 2 z 1 Pochhammer 1 2 n -1 1 2 2 z -1 z Erf z 1 2 LaguerreL n -1 -1 1 2 -1 z 2 n LaguerreL n -1 3 2 -1 z n z 1 2 -1 p 0 n -1 1 p 1 -1 LaguerreL n -1 p -1 p -1 1 2 -1 z LaguerreL p -1 p -1 1 2 z 1 2 z 1 2 -1 p 0 n -2 1 p 1 -1 LaguerreL n -1 p -2 p 1 2 -1 z LaguerreL p -1 p -1 1 2 z -1 -1 n z Gamma 1 2 -1 n LaguerreL n -1 1 2 -1 z n SuperPlus [/itex]

 Rule Form

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 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02