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 HermiteH

 http://functions.wolfram.com/05.01.06.0021.01

 Input Form

 HermiteH[n, z] \[Proportional] (-1)^Floor[n/2] (Floor[n/2] + (-1)^(n - 1)/2)! E^(z^2/2) 2^n z^(n - 2 Floor[n/2]) Sum[(Subscript[A, k] z^(2 k) Hypergeometric0F1Regularized[ (-1)^(n - 1)/2 + k + 1, (-z^2) (Floor[n/2] + ((-1)^(n - 1) + 2)/4)])/ 2^k, {k, 0, Infinity}] /; (n -> Infinity) && Subscript[A, 0] == 1 && Subscript[A, 1] == 0 && Subscript[A, 2] == ((-1)^(n - 1) + 2)/4 && Subscript[A, m] == ((2 m + (-1)^(n - 1) - 2)/(2 m)) Subscript[A, m - 2] - (2 Floor[n/2] + (-1)^(n - 1)/2 + 1) Subscript[A, m - 3] && Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "2"]]], ")"]], "!"]], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "2"]], SuperscriptBox["2", "n"], " ", SuperscriptBox["z", RowBox[List["n", "-", RowBox[List["2", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["A", "k"], SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List["2", "k"]]], RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "2"], "+", "k", "+", "1"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "+", "2"]], "4"]]], ")"]]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[SubscriptBox["A", "0"], "\[Equal]", "1"]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["A", "1"], "\[Equal]", "0"]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["A", "2"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "+", "2"]], "4"]]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["A", "m"], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "-", "2"]], RowBox[List["2", "m"]]], SubscriptBox["A", RowBox[List["m", "-", "2"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "2"], "+", "1"]], ")"]], SubscriptBox["A", RowBox[List["m", "-", "3"]]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 H n ( z ) ( - 1 ) n 2 ( n 2 + ( - 1 ) n - 1 2 ) ! z 2 2 2 n z n - 2 n 2 k = 0 A k 2 - k z 2 k 0 F ~ 1 ( ; k + ( - 1 ) n - 1 2 + 1 ; - z 2 ( ( - 1 ) n - 1 + 2 4 + n 2 ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "2"], " ", "+", "1"]], Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "+", "2"]], "4"], " ", "+", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]], ")"]]]], Hypergeometric0F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] /; ( n "\[Rule]" ) A 0 1 A 1 0 A 2 ( - 1 ) n - 1 + 2 4 A m 2 m + ( - 1 ) n - 1 - 2 2 m A m - 2 - ( 2 n 2 + ( - 1 ) n - 1 2 + 1 ) A m - 3 m + Condition Proportional HermiteH n z -1 n 2 -1 n 2 -1 -1 n -1 2 -1 z 2 2 -1 2 n z n -1 2 n 2 -1 k 0 Subscript A k 2 -1 k z 2 k Hypergeometric0F1Regularized k -1 n -1 2 -1 1 -1 z 2 -1 n -1 2 4 -1 n 2 -1 Rule n Subscript A 0 1 Subscript A 1 0 Subscript A 2 -1 n -1 2 4 -1 Subscript A m 2 m -1 n -1 -2 2 m -1 Subscript A m -2 -1 2 n 2 -1 -1 n -1 2 -1 1 Subscript A m -3 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]]]]]], ")"]], "!"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "2"]], " ", SuperscriptBox["2", "n"], " ", SuperscriptBox["z", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["A", "k"], " ", SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]]]], "+", "k", "+", "1"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "+", "2"]], ")"]]]]]], ")"]]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[SubscriptBox["A", "0"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["A", "1"], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["A", "2"], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "+", "2"]], ")"]]]]]], "&&", RowBox[List[SubscriptBox["A", "m"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "-", "2"]], ")"]], " ", SubscriptBox["A", RowBox[List["m", "-", "2"]]]]], RowBox[List["2", " ", "m"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]]]], "+", "1"]], ")"]], " ", SubscriptBox["A", RowBox[List["m", "-", "3"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2007-05-02