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 Hypergeometric1F1

 http://functions.wolfram.com/07.20.03.0127.01

 Input Form

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

 Standard Form

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 MathML Form

 1 F 1 ( n ; 1 2 - m ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["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[FractionBox["1", "2"], "-", "m"]], 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 ( m + 1 ) ! ) 2 ( m + n ) ! ( - 1 2 ) m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["m", "+", "1"]]], Pochhammer] ( π z erf ( z ) z k = 0 m + 1 ( k + m + n ) ! k ! L - k + m + 1 k - m - 1 2 ( z ) L k + m + n - k - 1 2 ( - z ) + k = 0 m + 1 ( k + m + n ) ! k ! L - k + m + 1 k - m - 1 2 ( z ) p = 1 k + m + n 1 p L k + m + n - p - k + p - 1 2 ( - z ) L p - 1 1 2 - p ( z ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n - m m Condition Hypergeometric1F1 n 1 2 -1 m z -1 m 1 m 1 2 m n Pochhammer -1 1 2 m 1 -1 1 2 z Erf z 1 2 z 1 2 -1 k 0 m 1 k m n k -1 LaguerreL -1 k m 1 k -1 m -1 1 2 z LaguerreL k m n -1 k -1 1 2 -1 z k 0 m 1 k m n k -1 LaguerreL -1 k m 1 k -1 m -1 1 2 z p 1 k m n 1 p -1 LaguerreL k m n -1 p -1 k p -1 1 2 -1 z LaguerreL p -1 1 2 -1 p z n n -1 m m [/itex]

 Rule Form

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

 2007-05-02