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 HermiteH

 http://functions.wolfram.com/05.01.21.0009.01

 Input Form

 Integrate[(z^(\[Alpha] - 1) HermiteH[n, z])/E^(p z^2), z] == n! Sum[(1/(k! (n - 2 k)!)) ((-1)^(k - 1) 2^(n - 2 k - 1) z^(n - 2 k + \[Alpha]) (p z^2)^(k - (1/2) (n + \[Alpha])) Gamma[(1/2) (n + \[Alpha]) - k, p z^2]), {k, 0, Floor[n/2]}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", SuperscriptBox["z", "2"]]]], RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", "k"]]]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", "k"]], "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["n", "-", RowBox[List["2", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["k", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", "\[Alpha]"]], ")"]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", "\[Alpha]"]], ")"]]]], "-", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]]]]]]]]]]]]

 MathML Form

 z α - 1 - p z 2 H n ( z ) z n ! k = 0 n 2 ( - 1 ) k - 1 2 n - 2 k - 1 z n - 2 k + α ( p z 2 ) k - n + α 2 Γ ( n + α 2 - k , p z 2 ) k ! ( n - 2 k ) ! z z α -1 -1 p z 2 HermiteH n z n k 0 n 2 -1 -1 k -1 2 n -1 2 k -1 z n -1 2 k α p z 2 k -1 n α 2 -1 Gamma n α 2 -1 -1 k p z 2 k n -1 2 k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p_"]], " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", RowBox[List["2", " ", "k"]], "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["n", "-", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["k", "-", FractionBox[RowBox[List["n", "+", "\[Alpha]"]], "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "+", "\[Alpha]"]], "2"], "-", "k"]], ",", RowBox[List["p", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]]]]]]]]]]

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

 2001-10-29