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 HypergeometricU

 http://functions.wolfram.com/07.33.20.0018.01

 Input Form

 Derivative[1, 0, 0][HypergeometricU][m, n, z] == (-(Log[z] + PolyGamma[m])) HypergeometricU[m, n, z] + (1/(m - 1)!) Sum[(Binomial[n - m - 1, k] (m + k - 1)! PolyGamma[m + k])/ z^(m + k), {k, 0, n - m - 1}] - (1/(m - 1)!) Sum[Binomial[n - m - 1, k] (k + m - 1)! (-z)^(-k - m) (Sum[(LaguerreL[k + m - 1 - p, -k - m + p, -z] LaguerreL[p - 1, -p, z])/ p, {p, 1, k + m - 1}] - E^z (-CoshIntegral[z] + SinhIntegral[z]) LaguerreL[k + m - 1, -k - m, -z]), {k, 0, n - m - 1}] /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > m

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0", ",", "0"]], "]"]], "[", "HypergeometricU", "]"]], "[", RowBox[List["m", ",", "n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "m", "]"]]]], ")"]]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List["m", ",", "n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "k"]], "]"]]]], SuperscriptBox["z", RowBox[List["m", "+", "k"]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["k", "+", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1", "-", "p"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m", "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["p", "-", "1"]], ",", RowBox[List["-", "p"]], ",", "z"]], "]"]]]], "p"]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["CoshIntegral", "[", "z", "]"]]]], "+", RowBox[List["SinhIntegral", "[", "z", "]"]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "m"]]]]]]]]

 MathML Form

 U TagBox["U", HypergeometricU] ( 1 , 0 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative] ( m , n , z ) - ( log ( z ) + ψ ( m ) ) U TagBox[RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["log", "(", "z", ")"]], "+", RowBox[List[TagBox["\[Psi]", PolyGamma], "(", "m", ")"]]]], ")"]], "U"]]]], HypergeometricU] ( m , n , z ) + 1 ( m - 1 ) ! k = 0 n - m - 1 ( n - m - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( k + m - 1 ) ! ψ TagBox["\[Psi]", PolyGamma] ( k + m ) z - k - m - 1 ( m - 1 ) ! k = 0 n - m - 1 ( n - m - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( k + m - 1 ) ! ( - z ) - k - m ( p = 1 k + m - 1 1 p L k + m - p - 1 p - k - m ( - z ) L p - 1 - p ( z ) - z ( Shi ( z ) - Chi ( z ) ) L k + m - 1 - k - m ( - z ) ) /; m TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > m U TagBox["U", HypergeometricU] ( 1 , 0 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative] ( m , n , z ) - ( log ( z ) + ψ ( m ) ) U TagBox[RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["log", "(", "z", ")"]], "+", RowBox[List[TagBox["\[Psi]", PolyGamma], "(", "m", ")"]]]], ")"]], "U"]]]], HypergeometricU] ( m , n , z ) + 1 ( m - 1 ) ! k = 0 n - m - 1 ( n - m - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( k + m - 1 ) ! ψ TagBox["\[Psi]", PolyGamma] ( k + m ) z - k - m - 1 ( m - 1 ) ! k = 0 n - m - 1 ( n - m - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( k + m - 1 ) ! ( - z ) - k - m ( p = 1 k + m - 1 1 p L k + m - p - 1 p - k - m ( - z ) L p - 1 - p ( z ) - z ( Shi ( z ) - Chi ( z ) ) L k + m - 1 - k - m ( - z ) ) /; m TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > m [/itex]

 Rule Form

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 Contributed by

 Brychkov Yu.A. (2007)

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

 2007-05-02