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variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Polynomials > HermiteH[n,z] > Generating functions





http://functions.wolfram.com/05.01.11.0004.01









  


  










Input Form





HermiteH[n, z] == SeriesTerm[((Floor[n/2]! Pochhammer[1/2, Floor[n/2]])/ Pochhammer[c, Floor[n/2]]) (Hypergeometric1F1[c, 1/2, (4 z^2 t^2)/(1 + 4 t^2)]/(1 + 4 t^2)^c + ((32 c t^3 z^3)/(3 (1 + 4 t^2)^(c + 2))) Hypergeometric1F1[c + 1, 5/2, (4 z^2 t^2)/(1 + 4 t^2)] + ((2 z t (1 + 4 t^2 - 8 c t^2))/ (1 + 4 t^2)^(c + 1)) Hypergeometric1F1[c, 3/2, (4 z^2 t^2)/(1 + 4 t^2)]), {t, 0, n}]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]]]], ")"]], RowBox[List["-", "c"]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List["c", ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", SuperscriptBox["t", "2"]]], RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["32", " ", "c", " ", SuperscriptBox["t", "3"], " ", SuperscriptBox["z", "3"]]], ")"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["c", "+", "1"]], ",", FractionBox["5", "2"], ",", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", SuperscriptBox["t", "2"]]], RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]]]]]]], "]"]]]], RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]]]], ")"]], RowBox[List["c", "+", "2"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z", " ", "t", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]], "-", RowBox[List["8", " ", "c", " ", SuperscriptBox["t", "2"]]]]], ")"]]]], ")"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List["c", ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", SuperscriptBox["t", "2"]]], RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["t", "2"]]]]], ")"]], RowBox[List["c", "+", "1"]]]]]], ")"]]]], RowBox[List["Pochhammer", "[", RowBox[List["c", ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.