Zernike polynomials

Remarks

Each $Z_n^{\pm m}$ is a polynomial in the two variables $x = \rho \cos \varphi$ and $y = \rho \sin \varphi$, of degree $n$. Examples: $Z_0^0 = 1$, $Z_1^1 = \rho \cos \varphi = x$ and $Z_2^0 = 2\rho^2 - 1 = 2x^2 + 2y^2 - 1$. Check 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 $-\frac{1!}{1! \, 0! \, 0!} = -1$. Orthogonality: $\int_0^1 R_n^m(\rho) R_{n'}^m(\rho) \, \rho \, d\rho = \frac{\delta_{n n'}}{2(n + 1)}$; for example $\int_0^1 (2\rho^2 - 1) \rho \, d\rho = \frac12 - \frac12 = 0$. The normalisation is $R_n^m(1) = 1$. The radial parts are Jacobi polynomials: $R_n^m(\rho) = (-1)^{(n - m)/2} \rho^m P_{(n - m)/2}^{(m, 0)}(1 - 2\rho^2)$. Non-example: $n = 1$, $m = 0$ is excluded because $n - m$ is odd. The natural candidate $\rho = \sqrt{x^2 + y^2}$ is rotation invariant but is not a polynomial in $x$ and $y$.