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 WhittakerW

 http://functions.wolfram.com/07.45.20.0011.01

 Input Form

 D[z^\[Alpha] WhittakerW[\[Nu], \[Mu], z], {z, n}] == Sum[((-1)^(Subscript[k, 1] + Subscript[k, 3]) KroneckerDelta[n, Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3]] Multinomial[Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]] z^(\[Alpha] - Subscript[k, 2] - Subscript[k, 3]/2) Pochhammer[1/2 + \[Mu] - \[Nu], Subscript[k, 3]] Pochhammer[3/2 + \[Alpha] + \[Mu] - Subscript[k, 2], Subscript[k, 2]] WhittakerW[\[Nu] - Subscript[k, 3]/2, \[Mu] + Subscript[k, 3]/2, z])/ 2^Subscript[k, 1], {Subscript[k, 1], 0, n}, {Subscript[k, 2], 0, n}, {Subscript[k, 3], 0, n}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "3"]]]], RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", SubscriptBox["k", "1"]]]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", SubscriptBox["k", "2"], "-", FractionBox[SubscriptBox["k", "3"], "2"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", SubscriptBox["k", "3"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Alpha]", "+", "\[Mu]", "-", SubscriptBox["k", "2"]]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox[SubscriptBox["k", "3"], "2"]]], ",", RowBox[List["\[Mu]", "+", FractionBox[SubscriptBox["k", "3"], "2"]]], ",", "z"]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n ( z α W TagBox["W", WhittakerW] ν , μ ( z ) ) z n k 1 = 0 n k 2 = 0 n k 3 = 0 n ( - 1 ) k 1 + k 3 δ KroneckerDelta n , k 1 + k 2 + k 3 ( k 1 + k 2 + k 3 ; k 1 , k 2 , k 3 ) TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]], ";", SubscriptBox["k", "1"]]], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], ")"]], Multinomial, Rule[Editable, False]] 2 - k 1 z α - k 2 - k 3 2 ( μ - ν + 1 2 ) k 3 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], SubscriptBox["k", "3"]], Pochhammer] ( α + μ - k 2 + 3 2 ) k 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "+", "\[Mu]", "-", SubscriptBox["k", "2"], "+", FractionBox["3", "2"]]], ")"]], SubscriptBox["k", "2"]], Pochhammer] W TagBox["W", WhittakerW] ν - k 3 2 , μ + k 3 2 ( z ) /; n Condition z n z α WhittakerW ν μ z Subscript k 3 0 n Subscript k 2 0 n Subscript k 1 0 n -1 Subscript k 1 Subscript k 3 KroneckerDelta n Subscript k 1 Subscript k 2 Subscript k 3 Multinomial Subscript k 1 Subscript k 2 Subscript k 3 2 -1 Subscript k 1 z α -1 Subscript k 2 -1 Subscript k 3 2 -1 Pochhammer μ -1 ν 1 2 Subscript k 3 Pochhammer α μ -1 Subscript k 2 3 2 Subscript k 2 WhittakerW ν -1 Subscript k 3 2 -1 μ Subscript k 3 2 -1 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z_", "\[Alpha]_"], " ", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "3"]]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["2", RowBox[List["-", SubscriptBox["k", "1"]]]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", SubscriptBox["k", "2"], "-", FractionBox[SubscriptBox["k", "3"], "2"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", SubscriptBox["k", "3"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Alpha]", "+", "\[Mu]", "-", SubscriptBox["k", "2"]]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox[SubscriptBox["k", "3"], "2"]]], ",", RowBox[List["\[Mu]", "+", FractionBox[SubscriptBox["k", "3"], "2"]]], ",", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02