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 WhittakerM

 http://functions.wolfram.com/07.44.03.0049.01

 Input Form

 WhittakerM[(2 m + 1)/4 + n, (2 m + 1)/4, z] == ((1/2) (-1)^m Sqrt[Pi] z^((1 - 2 m)/4) Erfi[Sqrt[z]] Pochhammer[3/2, m] Sum[Binomial[n, k] LaguerreL[k + m, -(1/2) - k - m, z], {k, 0, n}])/ E^(z/2) + (1/2) (-1)^m E^(z/2) z^((3 - 2 m)/4) Pochhammer[3/2, m] Sum[Binomial[n, k] Sum[(1/p) LaguerreL[k + m - p, -(1/2) - k - m + p, z] LaguerreL[-1 + p, 1/2 - p, -z], {p, 1, k + m}], {k, 0, n}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", "1"]], "4"], "+", "n"]], ",", FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", "1"]], "4"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], SuperscriptBox["z", FractionBox[RowBox[List["1", "-", RowBox[List["2", "m"]]]], "4"]], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "m"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k", "-", "m"]], ",", "z"]], "]"]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", SuperscriptBox["z", FractionBox[RowBox[List["3", "-", RowBox[List["2", "m"]]]], "4"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "m"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["k", "+", "m"]]], RowBox[List[FractionBox["1", "p"], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "p"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k", "-", "m", "+", "p"]], ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 M TagBox["M", WhittakerM] 2 m + 1 4 + n , 2 m + 1 4 ( z ) ( - 1 ) m π 2 ( 3 2 ) m TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "m"], Pochhammer] erfi ( z ) - z 2 z 1 - 2 m 4 k = 0 n ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] L k + m - k - m - 1 2 ( z ) + ( - 1 ) m 2 ( 3 2 ) m TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "m"], Pochhammer] z / 2 z 3 - 2 m 4 k = 0 n ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] p = 1 k + m 1 p L k + m - p - k - m + p - 1 2 ( z ) L p - 1 1 2 - p ( - z ) /; n m Condition WhittakerM 2 m 1 4 -1 n 2 m 1 4 -1 z -1 m 1 2 2 -1 Pochhammer 3 2 m Erfi z 1 2 -1 z 2 -1 z 1 -1 2 m 4 -1 k 0 n Binomial n k LaguerreL k m -1 k -1 m -1 1 2 z -1 m 2 -1 Pochhammer 3 2 m z 2 -1 z 3 -1 2 m 4 -1 k 0 n Binomial n k p 1 k m 1 p -1 LaguerreL k m -1 p -1 k -1 m p -1 1 2 z LaguerreL p -1 1 2 -1 p -1 z n m [/itex]

 Rule Form

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 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02