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variants of this functions
HermiteH




Mathematica Notation

Traditional Notation









Polynomials > HermiteH[n,z] > Representations through more general functions > Through other functions > Involving some hypergeometric-type functions





http://functions.wolfram.com/05.01.26.0015.01









  


  








Input Form





HermiteH[n, z] == Limit[(2^(n/2) Gamma[n + 1] LaguerreL[n, \[Lambda], \[Lambda] - Sqrt[2 \[Lambda]] z])/\[Lambda]^(n/2), \[Lambda] -> Infinity]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["n", "/", "2"]]], RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", SuperscriptBox["\[Lambda]", RowBox[List["-", FractionBox["n", "2"]]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["\[Lambda]", "-", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Lambda]"]]], " ", "z"]]]]]], "]"]]]], ",", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#955; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <limit /> <bvar> <ci> &#955; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#955; </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> &#955; </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["n", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", SuperscriptBox["\[Lambda]", RowBox[List["-", FractionBox["n", "2"]]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", RowBox[List["\[Lambda]", "-", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Lambda]"]]], " ", "z"]]]]]], "]"]]]], ",", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]


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





2001-10-29