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variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HermiteH[nu,z] > Differentiation > Low-order differentiation > With respect to nu





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





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MathML Form







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<mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <apply> <ci> Subscript </ci> <apply> <ci> BesselK </ci> <ci> H </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> 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</math>










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





© 1998- Wolfram Research, Inc.