Distances of roots of classical orthogonal polynomials

Michael Voit

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arXiv · 2109.14999

Distances of roots of classical orthogonal polynomials

Michael Voit

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Abstract

Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$ of $P_N$ we derive lower estimates for $\min_{i\ne j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality intervals. The proofs are based on recent results on the eigenvalues of the covariance matrices in central limit theorems for associated $\beta$-random matrix ensembles where these entities appear as entries, and where the eigenvalues of these matrices are known.

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Michael Voit. 2021-09-30. Distances of roots of classical orthogonal polynomials. https://arxiv.org/abs/2109.14999

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