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variants of this functions
ChebyshevT




Mathematica Notation

Traditional Notation









Polynomials > ChebyshevT[n,z] > Identities > Functional identities > Relations between contiguous functions > Normalized recurrence relation





http://functions.wolfram.com/05.04.17.0007.01









  


  








Input Form





z p[n, z] == p[n + 1, z] + (1/4) p[n - 1, z] /; p[n, z] == ChebyshevT[n, z]/2^n








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List["p", "[", RowBox[List["n", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "z"]], "]"]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List["p", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", "z"]], "]"]]]]]]]]]]








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> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <ci> z </ci> <apply> <ci> p </ci> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <ci> p </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> p </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> p </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["z_", " ", RowBox[List["p", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["p", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "z"]], "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["p", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["p", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", "z"]], "]"]]]]]]]]]]]]


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





2001-10-29