Symmetrization process and truncated orthogonal polynomials

2023/07/18 by Dominici, Diego, García-Ardila, Juan C., Marcellán, Francisco · 1 citation
#33C50 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2307.09581

Abstract

We define the family of truncated Laguerre polynomials P_n(x;z), orthogonal with respect to the linear functional ℓ defined by ⟨ℓ,p⟩=∫₀^zp(x)x^αe^-xdx, αgt;-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 Painlevé and Painlevé 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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