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 WhittakerM

 http://functions.wolfram.com/07.44.03.0052.01

 Input Form

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

 Standard Form

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 MathML Form

 M TagBox["M", WhittakerM] 2 m + 1 4 - n , 2 m - 1 4 ( z ) ( 2 m - 2 n + 1 ) ( m - n + 3 2 ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] 2 ( n - 1 ) ! - z 2 z 2 m + 1 4 ( z π z n - m - 1 2 erf ( z ) p = 0 n - 1 ( - z ) - p ( n - 1 p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["p", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 1 2 ) m - n + p TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["m", "-", "n", "+", "p"]]], Pochhammer] - 2 p = 0 n - 1 ( - 1 ) p ( n - 1 p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["p", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] 2 m - 2 n + 2 p + 1 k = 1 m - n + p ( - z ) - k ( n - m - p - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "-", "m", "-", "p", "-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ) /; n + m TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m n Condition WhittakerM 2 m 1 4 -1 -1 n 2 m -1 4 -1 z 2 m -1 2 n 1 Pochhammer m -1 n 3 2 n -1 2 n -1 -1 -1 z 2 -1 z 2 m 1 4 -1 z 1 2 z n -1 m -1 1 2 Erf z 1 2 p 0 n -1 -1 z -1 p Binomial n -1 p Pochhammer 1 2 m -1 n p -1 2 p 0 n -1 -1 p Binomial n -1 p 2 m -1 2 n 2 p 1 -1 k 1 m -1 n p -1 z -1 k Pochhammer n -1 m -1 p -1 1 2 k n SuperPlus m m n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m_"]], "+", "1"]], ")"]]]], "-", "n_"]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m_"]], "-", "1"]], ")"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "m"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "m", "-", "n"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "+", "1"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m", "+", "n"]]], " ", RowBox[List["Erf", "[", SqrtBox["z"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "p"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "p"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["m", "-", "n", "+", "p"]]]], "]"]]]]]]]], "-", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "p"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "n", "+", "p"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "m", "+", "n", "-", "p"]], ",", "k"]], "]"]]]]]]]], RowBox[List["1", "+", RowBox[List["2", " ", "m"]], "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "p"]]]]]]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "n"]]]]]]]]]]

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

 2007-05-02