On properties of a deformed Freud weight

Abstract

We study the recurrence coefficients of the monic polynomials P n (z) orthogonal with respect to the deformed (also called semi-classical) Freud weightwith parameters α > −1, N > 0, s ∈ [0, 1]. We show that the recurrence coefficients β n (s) satisfy the first discrete Painlevé equation (denoted by dP I ), a differential-difference equation and a second order nolinear ordinary differential equation (ODE) in s. Here n is the order of the Hankel matrix generated by w α (x; s, N ). We describe the asymptotic behavior of the recurrence coefficients in three situations, (i) s → 0, n, N finite, (ii) n → ∞, N finite, (iii) n, N → ∞, such that the radio r := n N is bounded away from 0 and closed to 1. We also investigate the existence and uniqueness for the positive solutions of the dP I .Further more, we derive, using the ladder approach, a second order linear ODE satisfied by the polynomials P n (z). It is found as n → ∞, the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, D n (s), associated with w α (x; s, N ) when n tends to infinity.

On semiclassical orthogonal polynomials associated with a Freud‐type weixght

“…See, for example, previous studies. 20,23,24 Remark 4.2. (40) is the well-known supplementary condition (S 1 ).…”

“…(40) is the well-known supplementary condition (S 1 ). 20,23,24 Lemma 4.3. The monic orthogonal polynomials {P n (z)} with respect to weight (2) satisfy the second-order ODE…”

“…In fact, Lemma 4.1 to 4.3 are not new results since many different problems can be obtained similar formulas, see, eg, previous studies. [24][25][26] However, we still give the proof for completeness and without assuming any reader's knowledge of these.…”

Laguerre–Hahn orthogonal polynomials on the real line

“…Usually, the deformation occurs via a t-dependence parameter on the weight (in many contexts, t is interpreted as a time variable), and the goal is to study the t-evolution of the recurrence coefficients of the orthogonal polynomials under such a dependence. Connections to random matrix theory, Lax pairs, Toda lattices, Painlevé equations, isomonodromic deformations, etc, are well-known and, indeed, intensively studied in the literature (see, amongst many others, [10,24,26,54,91]). Essentially, two methods of study have emerged: one has been to use a formulation in terms of Lax pairs for isomonodromic deformations of linear differential equations [54], and the other proceeds via the ladder operator technique for orthogonal polynomials [23].…”

Orthogonal Polynomials With a Semi-Classical Weight and Their Recurrence Coefficients

“…The study of orthogonal polynomials associated with various deformed weights appears in a vast list of articles. It has also been intensively studied by connection to random matrix theory, integrable systems, Painlevé equations and Heun equations, see [1]- [8], [16], [20], [29], [30]. A useful characterization of classical orthogonal polynomials is the Pearson equation…”