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arXiv · 2307.09581

Symmetrization process and truncated orthogonal polynomials

Diego Dominici·Juan C. García-Ardila·Francisco Marcellán
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Abstract

We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\ell$ defined by $$\langle{\ell,p\rangle}=\int_{0}^zp(x)x^\alpha e^{-x}dx,\qquad\alpha>-1.$$ The connection between $P_n(x;z)$ and the polynomials $S_n(x;z)$ (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials $P_n(x;z)$ and $S_n(x;z)$ are studied taking into account the relation between the parameters of the three-term recurrence relations that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlev\'e and Painlev\'e equations associated with such coefficients appear in a natural way. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameter $z$ are given.

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Diego Dominici, Juan C. García-Ardila, Francisco Marcellán. 2023-07-18. Symmetrization process and truncated orthogonal polynomials. https://arxiv.org/abs/2307.09581

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