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 HermiteH

 http://functions.wolfram.com/07.01.20.0011.01

 Input Form

 Derivative[1, 0][HermiteH][2 n, z] == 2^(2 n - 1) n! Pochhammer[1/2, n] ((1/(2 n)!) (-EulerGamma + PolyGamma[1/2 - n] + Pi Erfi[z] - 2 z^2 HypergeometricPFQ[{1, 1}, {3/2, 2}, z^2]) HermiteH[2 n, z] + (((-1)^n Sqrt[Pi])/Pochhammer[1/2, n]) E^z^2 Sum[(1/(1 + k)) (z LaguerreL[k, -(1/2) - k, -z^2] - ((Sqrt[Pi] z^(2 + 2 k))/k!) LaguerreL[-(1/2), 1/2 + k, -z^2]) LaguerreL[-1 - k + n, 1/2 + k, z^2], {k, 0, n - 1}] - (-1)^n Sum[(PolyGamma[1/2 - k] (-4)^k z^(2 k))/((n - k)! (2 k)!), {k, 0, n}]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "HermiteH", "]"]], "[", RowBox[List[RowBox[List["2", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", "n"]], "-", "1"]]], " ", RowBox[List["n", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Erfi", "[", "z", "]"]]]], "-", RowBox[List["2", SuperscriptBox["z", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2"]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]], RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["2", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SqrtBox["\[Pi]"]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]], SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["1", "+", "k"]]], RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["LaguerreL", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]]]]], RowBox[List["k", "!"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 H TagBox["H", BesselK] 2 n ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) 2 2 n - 1 n ! ( 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] ( 1 ( 2 n ) ! ( - 2 2 F 2 ( 1 , 1 ; 3 2 , 2 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[SuperscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] z 2 + π erfi ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - n ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) H 2 n ( z ) - ( - 1 ) n k = 0 n ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - k ) ( - 4 ) k z 2 k ( n - k ) ! ( 2 k ) ! + z 2 ( - 1 ) n π ( 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] k = 0 n - 1 1 k + 1 ( z L k - k - 1 2 ( - z 2 ) - π z 2 k + 2 k ! L - 1 2 k + 1 2 ( - z 2 ) ) L - k + n - 1 k + 1 2 ( z 2 ) ) /; n Condition 1 0 Subscript BesselK H 2 n z 2 2 n -1 n Pochhammer 1 2 n 1 2 n -1 -2 HypergeometricPFQ 1 1 3 2 2 z 2 z 2 Erfi z PolyGamma 1 2 -1 n -1 HermiteH 2 n z -1 -1 n k 0 n PolyGamma 1 2 -1 k -4 k z 2 k n -1 k 2 k -1 z 2 -1 n 1 2 Pochhammer 1 2 n -1 k 0 n -1 1 k 1 -1 z LaguerreL k -1 k -1 1 2 -1 z 2 -1 1 2 z 2 k 2 k -1 LaguerreL -1 1 2 k 1 2 -1 z 2 LaguerreL -1 k n -1 k 1 2 z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["HermiteH", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["2", " ", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Erfi", "[", "z", "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2"]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox["\[Pi]"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["LaguerreL", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List["k", "!"]]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["1", "+", "k"]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02