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variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Polynomials > HermiteH[n,z] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/05.01.17.0006.01









  


  










Input Form





HermiteH[n, z] == Subscript[\[ScriptCapitalC], m][n, z] HermiteH[n + m, z] - (1/(2 (m + n))) Subscript[\[ScriptCapitalC], m][n, z] HermiteH[n + m + 1, z] /; Subscript[\[ScriptCapitalC], 0][n, z] == 1 && Subscript[\[ScriptCapitalC], 1][n, z] == z/(n + 1) && Subscript[\[ScriptCapitalC], m][n, z] == (z/(m + n)) Subscript[\[ScriptCapitalC], m - 1][n, z] - (1/(2 (m - 1 + n))) Subscript[\[ScriptCapitalC], m - 2][n, z] && Element[m, Integers] && m > 0










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mi> z </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> m </ci> </apply> <ci> n </ci> <ci> z </ci> </apply> <apply> <ci> HermiteH </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> m </ci> </apply> <ci> n </ci> <ci> z </ci> </apply> <apply> <ci> HermiteH </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 0 </cn> </apply> <ci> n </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> m </ci> </apply> <ci> n </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> <ci> n </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </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[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "z"]], "]"]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["n", "+", "m"]], ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "z"]], "]"]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List["n", "+", "m", "+", "1"]], ",", "z"]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "+", "n"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", FractionBox["z", RowBox[List["n", "+", "1"]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["n", ",", "z"]], "]"]]]], RowBox[List["m", "+", "n"]]], "-", FractionBox[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "2"]]], "[", RowBox[List["n", ",", "z"]], "]"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "1", "+", "n"]], ")"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]










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





2007-05-02





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