Generalized Gauss-Rys orthogonal polynomials

2024/01/31 by García-Ardila, Juan C., Marcellán, Francisco
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.17674

Abstract

Let (P_n(x;z;λ))_{n≥ 0} be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional s defined by \langles,p⟩=∫_-1¹ p(x)(1-x²)^(λ-1/2) e^-zx2dx, λgt;-1/2, zgt;0. In this contribution, several properties of the polynomials P_n(x;z;λ) are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters z and λ are given.

Related

• Symmetrization process and truncated orthogonal polynomials
• Symmetric Truncated Freud polynomials
• Multiple orthogonal polynomials with respect to Gauss' hypergeometric function
• Orthogonal polynomials associated with an inverse quadratic spectral transform
• Preud's equations for orthogonal polynomials as discrete Painlevé equations