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.

中文翻译：

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限制在 Zernike 圆多项式（圆盘多项式）的径向导数上

我们在 $n$ 到 $n^2() 阶泽尼克圆多项式（圆盘多项式）的 $k^{{\rm th}}$ 径向导数的最大模数上锐化边界 $n^{2k}$ n^2-1^2)\cdot ... \cdot(n^2-(k-1)^2)/2^k(1/2)_k$。该界是从 Koornwinder 将圆多项式的径向部分的连接系数的非负性展开为一系列第一类 Chebyshev 多项式时的结果获得的。例如，当 $m=O(\sqrt{n})$ 时，通过使用连接系数的显式表达式，新的界限对于 $n$ 和方位角阶 $m$ 的 Zernike 圆多项式是尖锐的Jacobi 多项式的平方项为 0。 关键词：Zernike 圆多项式，圆盘多项式，径向导数，Chebyshev 多项式，连接系数，Gegenbauer 多项式。