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 HermiteH

 http://functions.wolfram.com/07.01.03.0014.01

 Input Form

 HermiteH[-n, z] == (-((I^(n + 1) Sqrt[Pi])/(2^n (n - 1)!))) E^z^2 HermiteH[n - 1, I z] - ((2^(-2 Floor[n/2] - 1) z^(1 - n + 2 Floor[n/2]))/ Pochhammer[1/2, Floor[n/2]]) E^z^2 Sum[((-1)^k/k!) LaguerreL[n - Floor[n/2] - k - 1, k, -z^2] (Gamma[-(1/2) + k + n - 2 Floor[n/2]] Erf[z] - ((-1)^(k + n) z Sum[(((-1)^j Gamma[-(1/2) - j])/ Gamma[3/2 - k - n + 2 Floor[n/2]]) z^(2 j), {j, 0, n - 2 Floor[n/2] + k - 2}])/E^z^2), {k, 0, n - Floor[n/2] - 1}] + (4^(-Floor[n/2] - 1)/Pochhammer[1/2, Floor[n/2]]) LaguerreL[n - Floor[n/2] - 1, -z^2] (4 (1 - n + 2 Floor[n/2]) + (-1)^n z^(2 - n + 2 Floor[n/2]) Sum[(((-1)^j Gamma[-(1/2) - j])/Gamma[3/2 - n + 2 Floor[n/2]]) z^(2 j), {j, 0, n - 2 Floor[n/2] - 2}]) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "+", "1"]]], SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SqrtBox["\[Pi]"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", "n", "+", RowBox[List["2", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "!"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "k", "-", "1"]], ",", "k", ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "+", "n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]], RowBox[List["Erf", "[", "z", "]"]]]], " ", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n"]]], "z", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "+", "k", "-", "2"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "k", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]], SuperscriptBox["z", RowBox[List["2", "j"]]]]]]]]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["4", RowBox[List[RowBox[List["-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "-", "1"]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "1"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["4", RowBox[List["(", RowBox[List["1", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]]]], "+", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox["z", RowBox[List["2", "-", "n", "+", RowBox[List["2", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "-", "2"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]], SuperscriptBox["z", RowBox[List["2", "j"]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 H - n ( z ) - n + 1 2 - n π ( n - 1 ) ! z 2 H n - 1 ( z ) - 2 - 2 n 2 - 1 z 2 n 2 - n + 1 ( 1 2 ) n 2 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]], Pochhammer] z 2 k = 0 n - n 2 - 1 ( - 1 ) k k ! L n - n 2 - k - 1 k ( - z 2 ) ( Γ ( k + n - 2 n 2 - 1 2 ) erf ( z ) - ( - 1 ) k + n z - z 2 j = 0 k + n - 2 n 2 - 2 ( - 1 ) j Γ ( - j - 1 2 ) z 2 j Γ ( 2 n 2 - n - k + 3 2 ) ) + 4 - n 2 - 1 ( 1 2 ) n 2 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]], Pochhammer] L n - n 2 - 1 ( - z 2 ) ( ( - 1 ) n z 2 n 2 - n + 2 j = 0 n - 2 n 2 - 2 ( - 1 ) j Γ ( - j - 1 2 ) z 2 j Γ ( 2 n 2 - n + 3 2 ) + 4 ( 2 n 2 - n + 1 ) ) /; n + Condition HermiteH -1 n z -1 n 1 2 -1 n 1 2 n -1 -1 z 2 HermiteH n -1 z -1 2 -2 n 2 -1 -1 z 2 n 2 -1 -1 n 1 Pochhammer 1 2 n 2 -1 -1 z 2 k 0 n -1 n 2 -1 -1 -1 k k -1 LaguerreL n -1 n 2 -1 -1 k -1 k -1 z 2 Gamma k n -1 2 n 2 -1 -1 1 2 Erf z -1 -1 k n z -1 z 2 j 0 k n -1 2 n 2 -1 -2 -1 j Gamma -1 j -1 1 2 z 2 j Gamma 2 n 2 -1 -1 n -1 k 3 2 -1 4 -1 n 2 -1 -1 Pochhammer 1 2 n 2 -1 -1 LaguerreL n -1 n 2 -1 -1 -1 z 2 -1 n z 2 n 2 -1 -1 n 2 j 0 n -1 2 n 2 -1 -2 -1 j Gamma -1 j -1 1 2 z 2 j Gamma 2 n 2 -1 -1 n 3 2 -1 4 2 n 2 -1 -1 n 1 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["-", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "+", "1"]]], " ", SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "k", "-", "1"]], ",", "k", ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "+", "n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]], " ", RowBox[List["Erf", "[", "z", "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n"]]], " ", "z", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "+", "k", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "j"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "k", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]]]]]]], ")"]]]], RowBox[List["k", "!"]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]], "+", FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "-", "1"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "-", "1"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", RowBox[List["2", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "-", "2"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "j"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "j"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]]]]]]], ")"]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02