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 HypergeometricU

 http://functions.wolfram.com/07.33.20.0017.01

 Input Form

 D[HypergeometricU[a, -n, z], {z, \[Alpha]}] == ((-1)^n/(z^\[Alpha] Gamma[a])) ((1/(n + 1)!) Sum[((Pochhammer[a + n + 1, k] z^(k + n + 1))/ (Pochhammer[n + 2, k] k!)) FDLogConstant[z, k + n + 1, \[Alpha]], {k, 0, Infinity}] - Sum[(Pochhammer[a, k] (-1)^(k + n - 1) (n - k)! z^k)/(Pochhammer[a, n + 1] Gamma[k - \[Alpha] + 1]), {k, 0, n}] + Sum[(Pochhammer[a + n + 1, k]/(Gamma[k + n + 2 - \[Alpha]] k!)) (PolyGamma[a + n + k + 1] - PolyGamma[1 + k] - PolyGamma[n + k + 2]) z^(k + n + 1), {k, 0, Infinity}]) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", RowBox[List["-", "n"]], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", "a", "]"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "n", "+", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["k", "+", "n", "+", "1"]]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "2"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List["k", "+", "n", "+", "1"]], ",", "\[Alpha]"]], "]"]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]]]]]]], "+", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "n", "+", "1"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "+", "n", "+", "2", "-", "\[Alpha]"]], "]"]], RowBox[List["k", "!"]], " "]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "n", "+", "k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "k", "+", "2"]], "]"]]]], ")"]], SuperscriptBox["z", RowBox[List["k", "+", "n", "+", "1"]]]]]]]]], " ", ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 α U TagBox["U", HypergeometricU] ( a , - n , z ) z α ( - 1 ) n z - α Γ ( a ) ( k = 0 ( a + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] ( ψ TagBox["\[Psi]", PolyGamma] ( a + k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + n + 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ) z k + n + 1 Γ ( k + n - α + 2 ) k ! + 1 ( n + 1 ) ! k = 0 ( a + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] z k + n + 1 ( n + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], "k"], Pochhammer] k ! ℱ𝒞 log ( α ) ( z , k + n + 1 ) - k = 0 n ( - 1 ) k + n - 1 ( a ) k ( n - k ) ! z k ( a ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] Γ ( k - α + 1 ) ) /; n + Condition z α HypergeometricU a -1 n z -1 n z -1 α Gamma a -1 k 0 Pochhammer a n 1 k PolyGamma a k n 1 -1 PolyGamma k n 2 -1 PolyGamma k 1 z k n 1 Gamma k n -1 α 2 k -1 1 n 1 -1 k 0 Pochhammer a n 1 k z k n 1 Pochhammer n 2 k k -1 Subscript ℱ𝒞 log α z k n 1 -1 k 0 n -1 k n -1 Subscript a k n -1 k z k Pochhammer a n 1 Gamma k -1 α 1 -1 n SuperPlus [/itex]

 Rule Form

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

 2001-10-29