Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp bound on the radial derivatives of the Zernike circle polynomials
  (disk polynomials)   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.

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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg