Sharp bound on the radial derivatives of the Zernike circle polynomials (disk polynomials)

Abstract

We sharpen the bound $n^{2k}$ on the maximum modulus of the $k^{{\rm th}}$ radial derivative of the Zernike circle polynomials (disk polynomials) of degree $n$ to $n^2(n^2-1^2)\cdot ... \cdot(n^2-(k-1)^2)/2^k(1/2)_k$. This bound is obtained from a result of Koornwinder on the non-negativity of connection coefficients of the radial parts of the circle polynomials when expanded into a series of Chebyshev polynomials of the first kind. The new bound is shown to be sharp for, for instance, Zernike circle polynomials of degree $n$ and azimuthal order $m$ when $m=O(\sqrt{n})$ by using an explicit expression for the connection coefficients in terms of squares of Jacobi polynomials evaluated at 0. Keywords: Zernike circle polynomial, disk polynomial, radial derivative, Chebyshev polynomial, connection coefficient, Gegenbauer polynomial.