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 WhittakerW

 http://functions.wolfram.com/07.45.20.0016.01

 Input Form

 D[WhittakerW[\[Nu], \[Mu], z], {z, \[Alpha]}] == (-(z^(1/2 - \[Mu] - \[Alpha])/Gamma[1/2 + \[Mu] - \[Nu]])) (((-1)^(2 \[Mu])/(-2 \[Mu])!) Sum[(Pochhammer[1/2 - \[Mu] - \[Nu], j]/ (j! (k - j)! Pochhammer[1 - 2 \[Mu], j])) (-(1/2))^(k - j) FDLogConstant[z, k - \[Mu] + 1/2, \[Alpha]] z^k, {k, 0, Infinity}, {j, 0, k}] - (z^(2 \[Mu])/Pochhammer[1/2 + \[Mu] - \[Nu], -2 \[Mu]]) Sum[(((-1)^(k - j) (j - k - 2 \[Mu] - 1)! Pochhammer[ 1/2 + \[Mu] - \[Nu], k - j])/((k - j)! j!)) (-(1/2))^j FDPowerConstant[z, k + \[Mu] + 1/2, \[Alpha]] z^k, {k, 0, -1 - 2 \[Mu]}, {j, 0, k}] + (1/Pochhammer[1/2 + \[Mu] - \[Nu], -2 \[Mu]]) Sum[(((-1)^(2 \[Mu] + j) (-(1/2))^(j + k + 1) j! Pochhammer[1/2 + \[Mu] - \[Nu], -1 - 2 \[Mu] - j] Gamma[k - \[Mu] + 3/2])/((-1 - 2 \[Mu] - j)! (j + k + 1)! Gamma[k - \[Mu] + 3/2 - \[Alpha]])) z^k, {k, 0, Infinity}, {j, 0, -1 - 2 \[Mu]}] - (-1)^(2 \[Mu]) Sum[(((-(1/2))^(k - j) Pochhammer[1/2 - \[Mu] - \[Nu], j] Gamma[k - \[Mu] + 3/2])/(j! (j - 2 \[Mu])! (k - j)! Gamma[k - \[Mu] + 3/2 - \[Alpha]])) (PolyGamma[1 + j] + PolyGamma[1 + j - 2 \[Mu]] - PolyGamma[1/2 + j - \[Mu] - \[Nu]]) z^k, {k, 0, Infinity}, {j, 0, k}]) /; Element[-2 \[Mu], Integers] && -2 \[Mu] > 0

 Standard Form

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RowBox[List[RowBox[List["(", RowBox[List["j", "-", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]], RowBox[List["(", " ", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "-", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "j", "-", "\[Mu]", "-", "\[Nu]"]], "]"]]]], ")"]], SuperscriptBox["z", "k"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Mu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Mu]"]], ">", "0"]]]]]]]]

 MathML Form

 α W TagBox["W", WhittakerW] ν , μ ( z ) z α - z 1 2 - α - μ Γ ( μ - ν + 1 2 ) ( ( - 1 ) 2 μ ( - 2 μ ) ! k = 0 j = 0 k ( 1 2 - μ - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( - 1 2 ) k - j j ! ( k - j ) ! ( 1 - 2 μ ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "j"], Pochhammer] ℱ𝒞 log ( α ) ( z , k - μ + 1 2 ) z k - z 2 μ ( μ - ν + 1 2 ) - 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], Pochhammer] k = 0 - 2 μ - 1 j = 0 k ( - 1 ) k - j ( j - k - 2 μ - 1 ) ! ( μ - ν + 1 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( - 1 2 ) j ( k - j ) ! j ! ℱ𝒞 exp ( α ) ( z , k + μ + 1 2 ) z k + 1 ( μ - ν + 1 2 ) - 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], Pochhammer] k = 0 j = 0 - 2 μ - 1 ( - 1 ) j + 2 μ ( - 1 2 ) j + k + 1 j ! ( μ - ν + 1 2 ) - j - 2 μ - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "-", RowBox[List["2", " ", "\[Mu]"]], "-", "1"]]], Pochhammer] Γ ( k - μ + 3 2 ) z k ( - j - 2 μ - 1 ) ! ( j + k + 1 ) ! Γ ( k - α - μ + 3 2 ) - ( - 1 ) 2 μ k = 0 j = 0 k ( - 1 2 ) k - j ( 1 2 - μ - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] Γ ( k - μ + 3 2 ) j ! ( j - 2 μ ) ! ( k - j ) ! Γ ( k - α - μ + 3 2 ) ( ψ TagBox["\[Psi]", PolyGamma] ( j + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( j - 2 μ + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j - μ - ν + 1 2 ) ) z k ) /; - 2 μ + Condition z α WhittakerW ν μ z -1 z 1 2 -1 α -1 μ Gamma μ -1 ν 1 2 -1 -1 2 μ -2 μ -1 j 0 k k 0 Pochhammer 1 2 -1 μ -1 ν j -1 1 2 k -1 j j k -1 j Pochhammer 1 -1 2 μ j -1 Subscript ℱ𝒞 log α z k -1 μ 1 2 z k -1 z 2 μ Pochhammer μ -1 ν 1 2 -2 μ -1 j 0 k k 0 -2 μ -1 -1 k -1 j j -1 k -1 2 μ -1 Pochhammer μ -1 ν 1 2 k -1 j -1 1 2 j k -1 j j -1 Subscript ℱ𝒞 exp α z k μ 1 2 z k 1 Pochhammer μ -1 ν 1 2 -2 μ -1 j 0 -2 μ -1 k 0 -1 j 2 μ -1 1 2 j k 1 j Pochhammer μ -1 ν 1 2 -1 j -1 2 μ -1 Gamma k -1 μ 3 2 z k -1 j -1 2 μ -1 j k 1 Gamma k -1 α -1 μ 3 2 -1 -1 -1 2 μ j 0 k k 0 -1 1 2 k -1 j Pochhammer 1 2 -1 μ -1 ν j Gamma k -1 μ 3 2 j j -1 2 μ k -1 j Gamma k -1 α -1 μ 3 2 -1 PolyGamma j 1 PolyGamma j -1 2 μ 1 -1 PolyGamma j -1 μ -1 ν 1 2 z k -2 μ SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], 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RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Mu]", "+", FractionBox["3", "2"], "-", "\[Alpha]"]], "]"]]]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]]], "]"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["2", " ", "\[Mu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02