Chebyshev Expansion of the Zernike Polynomials
Richard J. Mathar
TL;DR
This work addresses numerical instability in evaluating Zernike radial polynomials by re-expressing $R_n^m(x)$ in a Chebyshev basis: $R_n^m(x)=\sum'_i c_{n,m,i} T_i(x)$. It derives closed-form rational coefficients for the small-gap regime $0\le n-m\le 6$ and establishes a holonomic five-term recurrence to compute $c_{n,m,i}$ for larger $n$, with clear parity selection rules $c_{n,m,i}=0$ when $n-m$ or $n-i$ is odd. The paper also extends the framework to higher dimensions, defining $R_n^{(m)}(x)$ in dimension $D\ge 2$ and introducing $c_{D,n,m,i}$ with a generalized ${}_4F_3$ representation and a dimension-dependent five-term recurrence, preserving orthogonality under $x^{D-1}$ weight. Collectively, these results enable stable, exact computation of Zernike polynomials in both circle ($D=2$) and higher-dimensional settings, with accompanying software and tables for practical use.
Abstract
The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansion coefficients c_{n,m,i} for a set of small 0 <= n-m <= 6 and a holonomic five-term recurrence for these coefficients for all larger n.