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http://functions.wolfram.com/07.01.21.0009.01
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Integrate[(z^(\[Alpha] - 1) HermiteH[\[Nu], a z])/E^(a^2 z^2), z] ==
2^(-1 + \[Nu]) Pi z^\[Alpha] ((1/Gamma[(1 - \[Nu])/2])
(Gamma[\[Alpha]/2] HypergeometricPFQRegularized[
{(1 + \[Nu])/2, \[Alpha]/2}, {1/2, (2 + \[Alpha])/2}, (-a^2) z^2]) -
(1/Gamma[-(\[Nu]/2)]) (a z Gamma[(1 + \[Alpha])/2]
HypergeometricPFQRegularized[{(2 + \[Nu])/2, (1 + \[Alpha])/2},
{3/2, (3 + \[Alpha])/2}, (-a^2) z^2]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", "\[Pi]", " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", FractionBox["\[Alpha]", "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox["\[Alpha]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["2", "+", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]], RowBox[List["(", RowBox[List["a", " ", "z", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]
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Date Added to functions.wolfram.com (modification date)
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