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HypergeometricU




Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricU[a,b,z] > Differentiation > Low-order differentiation > Mixed derivatives by parameters





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







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type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










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