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 HermiteH

 http://functions.wolfram.com/05.01.21.0021.01

 Input Form

 Integrate[(HermiteH[n, z]^2 Log[HermiteH[n, z]^2])/E^z^2, {z, -Infinity, Infinity}] == 2^n n! Sqrt[Pi] Log[2^(2 n)] - 2 Sum[Subscript[V, n][Subscript[z, k]], {k, 1, n}] /; HermiteH[n, Subscript[z, k]] == 0 && Subscript[V, n][z] == 2^n n! Sqrt[Pi] (EulerGamma/2 + Log[2] - z^2 HypergeometricPFQ[{1, 1}, {3/2, 2}, -z^2] + (1/2) Sum[(((-1)^k 2^k)/k) Binomial[n, k] Hypergeometric1F1[k, 1/2, -z^2], {k, 1, n}])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "2"], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "2"], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Log", "[", SuperscriptBox["2", RowBox[List["2", " ", "n"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SubscriptBox["V", "n"], "[", SubscriptBox["z", "k"], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", SubscriptBox["z", "k"]]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["V", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox["EulerGamma", "2"], "+", RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2"]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", "k"]]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List["k", ",", FractionBox["1", "2"], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]

 MathML Form

 - H n ( z ) 2 log ( H n ( z ) 2 ) - z 2 z 2 n n ! π log ( 2 2 n ) - 2 k = 1 n V n ( z k ) /; H n ( z k ) 0 V n ( z ) 2 n n ! π ( log ( 2 ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] 2 - 2 F 2 ( 1 , 1 ; 3 2 , 2 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] z 2 + 1 2 k = 1 n ( - 1 ) k 2 k k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 1 F 1 ( k ; 1 2 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["k", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] ) Condition z -1 HermiteH n z 2 HermiteH n z 2 -1 z 2 2 n n 1 2 2 2 n -1 2 k 1 n Subscript V n Subscript z k HermiteH n Subscript z k 0 Subscript V n z 2 n n 1 2 2 2 -1 -1 HypergeometricPFQ 1 1 3 2 2 -1 z 2 z 2 1 2 k 1 n -1 k 2 k k -1 Binomial n k Hypergeometric1F1 k 1 2 -1 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "2"], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "2"], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z_", "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Log", "[", SuperscriptBox["2", RowBox[List["2", " ", "n"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SubscriptBox["V", "n"], "[", SubscriptBox["zz", "k"], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", SubscriptBox["zz", "k"]]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List[SubscriptBox["V", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[FractionBox["EulerGamma", "2"], "+", RowBox[List["Log", "[", "2", "]"]], "-", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2"]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", "k"]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List["k", ",", FractionBox["1", "2"], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "k"]]]]]]], ")"]]]]]]]]]]]]]]

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

 2002-12-18