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 HermiteH

 http://functions.wolfram.com/07.01.21.0015.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) HermiteH[\[Nu], t])/E^(a t^2), {t, 0, Infinity}] == (2^(-1 + \[Nu]) Sqrt[Pi] ((Gamma[\[Alpha]/2] Hypergeometric2F1[\[Alpha]/2, -(\[Nu]/2), 1/2, 1/a])/ Gamma[(1 - \[Nu])/2] + (Gamma[(1 + \[Alpha])/2]/ (Sqrt[a] \[Alpha] Gamma[1 - \[Nu]/2])) ((-1 + a) Hypergeometric2F1[(1 + \[Alpha])/2, (1 - \[Nu])/2, -(1/2), 1/a] - (-2 + a - \[Alpha] + \[Nu]) Hypergeometric2F1[ (1 + \[Alpha])/2, (1 - \[Nu])/2, 1/2, 1/a])))/a^(\[Alpha]/2) /; Re[a] > 0 && Re[\[Alpha]] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[1]]], " ", RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["t", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox["\[Alpha]", "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "a"]]], "]"]]]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], "+", RowBox[List[FractionBox[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], RowBox[List[SqrtBox["a"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "a"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "a"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]]]]]]]]

 MathML Form

 0 t α - 1 - a t 2 H ν ( t ) t 2 ν - 1 a - α 2 π ( Γ ( α 2 ) Γ ( 1 - ν 2 ) 2 F 1 ( α 2 , - ν 2 ; 1 2 ; 1 a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["\[Alpha]", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox["1", "a"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] + Γ ( α + 1 2 ) a α Γ ( 1 - ν 2 ) ( ( a - 1 ) 2 F 1 ( α + 1 2 , 1 - ν 2 ; - 1 2 ; 1 a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox["1", "a"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] - ( a - α + ν - 2 ) 2 F 1 ( α + 1 2 , 1 - ν 2 ; 1 2 ; 1 a ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox["1", "a"], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] ) ) /; Re ( a ) > 0 Re ( α ) > 0 Condition t 0 DirectedInfinity 1 t α -1 -1 a t 2 HermiteH ν t 2 ν -1 a -1 α 2 -1 1 2 Gamma α 2 -1 Gamma 1 -1 ν 2 -1 -1 Hypergeometric2F1 α 2 -1 -1 ν 2 -1 1 2 1 a -1 Gamma α 1 2 -1 a 1 2 α Gamma 1 -1 ν 2 -1 -1 a -1 Hypergeometric2F1 α 1 2 -1 1 -1 ν 2 -1 -1 1 2 1 a -1 -1 a -1 α ν -2 Hypergeometric2F1 α 1 2 -1 1 -1 ν 2 -1 1 2 1 a -1 a 0 α 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a_"]], " ", SuperscriptBox["t_", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["a", RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox["\[Alpha]", "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "a"]]], "]"]]]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "a"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "-", "\[Alpha]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "a"]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["a"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], "]"]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]]]]]]]]]]

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

 2001-10-29