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http://functions.wolfram.com/05.09.25.0003.01
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Limit[GegenbauerC[n, \[Lambda], z/Sqrt[\[Lambda]]]/\[Lambda]^(n/2),
\[Lambda] -> Infinity] == (2/n!) HermiteH[n, z] /; Abs[z] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["\[Lambda]", RowBox[List["-", FractionBox["n", "2"]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", FractionBox[RowBox[List["z", " "]], SqrtBox["\[Lambda]"]]]], "]"]]]], ",", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["2", RowBox[List["n", "!"]]], RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> λ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msup> <mi> λ </mi> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mi> n </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mfrac> <mi> z </mi> <msqrt> <mi> λ </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> λ </ci> </bvar> <condition> <apply> <tendsto /> <ci> λ </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <ci> λ </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> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> λ </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> λ </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["\[Lambda]_", RowBox[List["-", FractionBox["n_", "2"]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", FractionBox["z_", SqrtBox["\[Lambda]_"]]]], "]"]]]], ",", RowBox[List["\[Lambda]_", "\[Rule]", "\[Infinity]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]]]], RowBox[List["n", "!"]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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