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 HypergeometricU

 http://functions.wolfram.com/07.33.20.0020.01

 Input Form

 Derivative[1, 0, 0][HypergeometricU][m, 2 m + n, z] + 2 Derivative[0, 1, 0][HypergeometricU][m, 2 m + n, z] == (-HypergeometricU[m, 2 m + n, z]) (Log[z] + PolyGamma[m]) + (1/(m - 1)!) Sum[(Binomial[m + n - 1, k] (k + m - 1)! PolyGamma[k + m])/ z^(k + m), {k, 0, m + n - 1}] + (1/(m - 1)!) Sum[Binomial[m + n - 1, k] (k + m - 1)! (-z)^(-k - m) (Sum[(LaguerreL[j - 1, -j, z] LaguerreL[k + m - j - 1, -k - m + j, -z])/ j, {j, 1, k + m - 1}] - E^z (SinhIntegral[z] - CoshIntegral[z]) LaguerreL[k + m - 1, -k - m, -z]), {k, 0, m + n - 1}] /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 U TagBox["U", HypergeometricU] ( 1 , 0 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative] ( m , 2 m + n , z ) + 2 U TagBox["U", HypergeometricU] ( 0 , 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]], Derivative] ( m , 2 m + n , z ) - U TagBox["U", HypergeometricU] ( m , 2 m + n , z ) ( log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( m ) ) + 1 ( m - 1 ) ! k = 0 m + n - 1 ( m + n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "+", "n", "-", "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 m + n - 1 ( m + n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "+", "n", "-", "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 ( j = 1 k + m - 1 1 j L j - 1 - j ( z ) L k + m - j - 1 j - k - m ( - z ) - z ( Shi ( z ) - Chi ( z ) ) L k + m - 1 - k - m ( - z ) ) /; m TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 0 HypergeometricU m 2 m n z 2 0 1 0 HypergeometricU m 2 m n z -1 HypergeometricU m 2 m n z z PolyGamma m 1 m -1 -1 k 0 m n -1 Binomial m n -1 k k m -1 PolyGamma k m z k m -1 1 m -1 -1 k 0 m n -1 Binomial m n -1 k k m -1 -1 z -1 k -1 m j 1 k m -1 1 j -1 LaguerreL j -1 -1 j z LaguerreL k m -1 j -1 j -1 k -1 m -1 z -1 z SinhIntegral z -1 CoshIntegral z LaguerreL k m -1 -1 k -1 m -1 z m SuperPlus n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["HypergeometricU", TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["m_", ",", RowBox[List[RowBox[List["2", " ", "m_"]], "+", "n_"]], ",", "z_"]], "]"]], "+", RowBox[List["2", " ", RowBox[List[SuperscriptBox["HypergeometricU", TagBox[RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["m_", ",", RowBox[List[RowBox[List["2", " ", "m_"]], "+", "n_"]], ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["HypergeometricU", "[", RowBox[List["m", ",", RowBox[List[RowBox[List["2", " ", "m"]], "+", "n"]], ",", "z"]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "m", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "+", "n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "+", "n", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "1"]], ")"]], "!"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "m"]], "]"]]]], SuperscriptBox["z", RowBox[List["k", "+", "m"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "+", "n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "+", "n", "-", "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["j", "=", "1"]], RowBox[List["k", "+", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["j", "-", "1"]], ",", RowBox[List["-", "j"]], ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "j", "-", "1"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m", "+", "j"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "j"]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List[RowBox[List["SinhIntegral", "[", "z", "]"]], "-", RowBox[List["CoshIntegral", "[", "z", "]"]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2007)

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

 2007-05-02

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