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ZernikeR






Mathematica Notation

Traditional Notation









Polynomials > ZernikeR[n,m,z] > Identities > Recurrence identities > Consecutive neighbors





http://functions.wolfram.com/05.18.17.0002.01









  


  










Input Form





ZernikeR[n, m, z] == (n/(-m^2 + n^2)) ((-((-m + n)^2/n) - (-2 + m + n)^2/(-2 + n) + 4 (-1 + n) z^2) ZernikeR[-2 + n, m, z] - ((-m^2 + (-2 + n)^2)/(-2 + n)) ZernikeR[-4 + n, m, z]) /; Element[n, Integers] && n >= 4 && Element[m, Integers] && m >= 0 && m <= n - 1










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> <msubsup> <mi> R </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mi> n </mi> </mfrac> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> R </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mn> 4 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8804; </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ZernikeR </ci> <ci> n </ci> <ci> m </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> ZernikeR </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <ci> m </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ZernikeR </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -4 </cn> </apply> <ci> m </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <geq /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <leq /> <ci> m </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ZernikeR", "[", RowBox[List["n_", ",", "m_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ")"]], "2"], "n"]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m", "+", "n"]], ")"]], "2"], RowBox[List[RowBox[List["-", "2"]], "+", "n"]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ",", "m", ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["m", "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n"]], ")"]], "2"]]], ")"]], " ", RowBox[List["ZernikeR", "[", RowBox[List[RowBox[List[RowBox[List["-", "4"]], "+", "n"]], ",", "m", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["-", "2"]], "+", "n"]]]]], ")"]]]], RowBox[List[RowBox[List["-", SuperscriptBox["m", "2"]]], "+", SuperscriptBox["n", "2"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "4"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[LessEqual]", RowBox[List["n", "-", "1"]]]]]]]]]]]]










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





2007-05-02





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