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  "body": "\n## Definition\n\nLet $n$ and $m$ be integers with $0 \\le m \\le n$ and $n - m$ even.\nThe *radial polynomial* $R_n^m$ on $[0, 1]$ is\n$$R_n^m(\\rho) = \\sum_{k=0}^{(n - m)/2} \\frac{(-1)^k \\, (n - k)!}{k! \\, \\left( \\frac{n + m}{2} - k \\right)! \\, \\left( \\frac{n - m}{2} - k \\right)!} \\, \\rho^{n - 2k}.$$\nThe *Zernike polynomials* are the functions on the closed unit disk, written in polar coordinates $(\\rho, \\varphi)$ with $0 \\le \\rho \\le 1$, given by\n$$Z_n^m(\\rho, \\varphi) = R_n^m(\\rho) \\cos(m\\varphi),$$\nand, for $m \u003e 0$,\n$$Z_n^{-m}(\\rho, \\varphi) = R_n^m(\\rho) \\sin(m\\varphi).$$\n\n## Remarks\n\nEach $Z_n^{\\pm m}$ is a [[polynomial in the two variables|07GNX8GCYP]] $x = \\rho \\cos \\varphi$ and $y = \\rho \\sin \\varphi$, of degree $n$.\nExamples: $Z_0^0 = 1$, $Z_1^1 = \\rho \\cos \\varphi = x$ and $Z_2^0 = 2\\rho^2 - 1 = 2x^2 + 2y^2 - 1$.\nCheck for $R_2^0$: the term $k = 0$ is $\\frac{2!}{0! \\, 1! \\, 1!} \\rho^2 = 2\\rho^2$ and the term $k = 1$ is $-\\frac{1!}{1! \\, 0! \\, 0!} = -1$.\nOrthogonality: $\\int_0^1 R_n^m(\\rho) R_{n'}^m(\\rho) \\, \\rho \\, d\\rho = \\frac{\\delta_{n n'}}{2(n + 1)}$; for example $\\int_0^1 (2\\rho^2 - 1) \\rho \\, d\\rho = \\frac12 - \\frac12 = 0$.\nThe normalisation is $R_n^m(1) = 1$.\nThe radial parts are Jacobi polynomials: $R_n^m(\\rho) = (-1)^{(n - m)/2} \\rho^m P_{(n - m)/2}^{(m, 0)}(1 - 2\\rho^2)$.\nNon-example: $n = 1$, $m = 0$ is excluded because $n - m$ is odd.\nThe natural candidate $\\rho = \\sqrt{x^2 + y^2}$ is rotation invariant but is not a polynomial in $x$ and $y$.\n",
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