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 WhittakerW

 http://functions.wolfram.com/07.45.20.0008.01

 Input Form

 Derivative[n, 0, 0][WhittakerW][\[Nu], \[Mu], z] == (Gamma[-2 \[Mu]] z^(\[Mu] + 1/2) Sum[D[Pochhammer[\[Mu] - \[Nu] + 1/2, k]/Gamma[1/2 - \[Mu] - \[Nu]], {\[Nu], n}] (z^k/(Pochhammer[2 \[Mu] + 1, k] k!)), {k, 0, Infinity}])/ E^(z/2) + (Gamma[2 \[Mu]] z^(1/2 - \[Mu]) Sum[D[Pochhammer[1/2 - \[Mu] - \[Nu], k]/Gamma[1/2 + \[Mu] - \[Nu]], {\[Nu], n}] (z^k/(Pochhammer[1 - 2 \[Mu], k] k!)), {k, 0, Infinity}])/ E^(z/2) /; Element[n, Integers] && n >= 0 && !Element[2 \[Mu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["n", ",", "0", ",", "0"]], "]"]], "[", "WhittakerW", "]"]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["D", "[", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]]], "]"]], FractionBox[SuperscriptBox["z", "k"], RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]]]]]], "+", " ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["D", "[", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]], ",", RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]]], "]"]], FractionBox[SuperscriptBox["z", "k"], RowBox[List[" ", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "\[Mu]"]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 W TagBox["W", BesselJ] ν , μ ( n , 0 , 0 ) TagBox[RowBox[List["(", RowBox[List["n", ",", "0", ",", "0"]], ")"]], Derivative] ( z ) - z 2 Γ ( - 2 μ ) z μ + 1 2 k = 0 n ( μ - ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] Γ ( 1 2 - μ - ν ) ν n z k ( 2 μ + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], "k"], Pochhammer] k ! + - z 2 Γ ( 2 μ ) z 1 2 - μ k = 0 n ( 1 2 - μ - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] Γ ( μ - ν + 1 2 ) ν n z k ( 1 - 2 μ ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "k"], Pochhammer] k ! /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] 2 μ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition n 0 0 Subscript BesselJ W ν μ z -1 z 2 -1 Gamma -2 μ z μ 1 2 k 0 ν n Pochhammer μ -1 ν 1 2 k Gamma 1 2 -1 μ -1 ν -1 z k Pochhammer 2 μ 1 k k -1 -1 z 2 -1 Gamma 2 μ z 1 2 -1 μ k 0 ν n Pochhammer 1 2 -1 μ -1 ν k Gamma μ -1 ν 1 2 -1 z k Pochhammer 1 -1 2 μ k k -1 n 2 μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["WhittakerW", TagBox[RowBox[List["(", RowBox[List["n", ",", "0", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]]]], FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Mu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", "n"]], "}"]]]]], FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "\[Element]", "Integers"]]]]]]]]]]]]

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

 2007-05-02