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 Hypergeometric1F1

 http://functions.wolfram.com/07.20.03.0139.01

 Input Form

 Hypergeometric1F1[1/2 - n, 1/2 + m, z] == (((-1)^(-1 + m) Sqrt[Pi])/2) z^(1/2 - m) Erfi[Sqrt[z]] Pochhammer[3/2, -1 + m] Sum[Binomial[n, k] LaguerreL[-1 + k + m, 1/2 - k - m, z], {k, 0, n}] + (1/2) (-1)^(-1 + m) E^z z^(1 - m) Pochhammer[3/2, -1 + m] Sum[Binomial[n, k] Sum[(1/p) LaguerreL[-1 + k + m - p, 1/2 - k - m + p, z] LaguerreL[-1 + p, 1/2 - p, -z], {p, 1, -1 + k + m}], {k, 0, n}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 1 F 1 ( 1 2 - n ; m + 1 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "n"]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["m", "+", FractionBox["1", "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric1F1] ( - 1 ) m - 1 2 ( 3 2 ) m - 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["m", "-", "1"]]], Pochhammer] z z 1 - m 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 1 p L k + m - p - 1 - k - m + p + 1 2 ( z ) L p - 1 1 2 - p ( - z ) + ( - 1 ) m - 1 π 2 ( 3 2 ) m - 1 z 1 2 - m erfi ( z ) 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 - 1 - k - m + 1 2 ( z ) /; n m + Condition Hypergeometric1F1 1 2 -1 n m 1 2 z -1 m -1 2 -1 Pochhammer 3 2 m -1 z z 1 -1 m k 0 n Binomial n k p 1 k m -1 1 p -1 LaguerreL k m -1 p -1 -1 k -1 m p 1 2 z LaguerreL p -1 1 2 -1 p -1 z -1 m -1 1 2 2 -1 Subscript 3 2 m -1 z 1 2 -1 m Erfi z 1 2 k 0 n Binomial n k LaguerreL k m -1 -1 k -1 m 1 2 z n m SuperPlus [/itex]

 Rule Form

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 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02