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arXiv · 1507.03749

Diophantine properties of the zeros of (monic) polynomials the coefficients of which are the zeros of Hermite polynomials

Oksana Bihun·Francesco Calogero
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Abstract

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of the polynomial p_N(z). We prove that the eigenvalues of M_1 and M_2 are the first N integers respectively the first N squared-integers, a remarkable isospectral and Diophantine property. The technique whereby these findings are demonstrated can be extended to other named polynomials.

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Oksana Bihun, Francesco Calogero. 2015-07-14. Diophantine properties of the zeros of (monic) polynomials the coefficients of which are the zeros of Hermite polynomials. https://arxiv.org/abs/1507.03749

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