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 HermiteH

 http://functions.wolfram.com/07.01.21.0007.01

 Input Form

 Integrate[(z^(\[Alpha] - 1) HermiteH[\[Nu], z])/E^(p z), z] == 2^\[Nu] Sqrt[Pi] ((-(z^\[Alpha]/(Gamma[(1 - \[Nu])/2] (p z)^\[Alpha]))) Sum[(Pochhammer[-(\[Nu]/2), k]/(Pochhammer[1/2, k] k! p^(2 k))) Gamma[2 k + \[Alpha], p z], {k, 0, Infinity}] + ((2 z^(1 + \[Alpha]))/(Gamma[-(\[Nu]/2)] (p z)^(\[Alpha] + 1))) Sum[(Pochhammer[(1 - \[Nu])/2, k]/(Pochhammer[3/2, k] k! p^(2 k))) Gamma[2 k + \[Alpha] + 1, p z], {k, 0, Infinity}])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", "z"]]], RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", "\[Alpha]"], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", "z"]], ")"]], "\[Alpha]"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], SuperscriptBox["p", RowBox[List["2", " ", "k"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List["p", " ", "z"]]]], "]"]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", "z"]], ")"]], RowBox[List["\[Alpha]", "+", "1"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], SuperscriptBox["p", RowBox[List["2", " ", "k"]]]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Alpha]", "+", "1"]], ",", RowBox[List["p", " ", "z"]]]], "]"]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 - p z H ν ( z ) z 2 ν π ( 2 z α + 1 Γ ( - ν 2 ) ( p z ) α + 1 k = 0 ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] Γ ( 2 k + α + 1 , p z ) ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! p 2 k - z α Γ ( 1 - ν 2 ) ( p z ) α k = 0 ( - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] Γ ( 2 k + α , p z ) ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! p 2 k ) z z α -1 -1 p z HermiteH ν z 2 ν 1 2 2 z α 1 Gamma -1 ν 2 -1 p z α 1 -1 k 0 Pochhammer 1 -1 ν 2 -1 k Gamma 2 k α 1 p z Pochhammer 3 2 k k p 2 k -1 -1 z α Gamma 1 -1 ν 2 -1 p z α -1 k 0 Pochhammer -1 ν 2 -1 k Gamma 2 k α p z Pochhammer 1 2 k k p 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_"]], " ", "z_"]]], " ", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List["p", " ", "z"]]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["p", RowBox[List["2", " ", "k"]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", "z"]], ")"]], "\[Alpha]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Alpha]", "+", "1"]], ",", RowBox[List["p", " ", "z"]]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["p", RowBox[List["2", " ", "k"]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", " ", "z"]], ")"]], RowBox[List["\[Alpha]", "+", "1"]]]]]]]], ")"]]]]]]]]

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

 2001-10-29