On semiclassical orthogonal polynomials associated with a Freud‐type weixght

Abstract

The recursion relationship: zP n (z) = P n+1 (z) + n P n−1 (z), n = 0, 1, 2 … is satisfied by all monic orthogonal polynomials in regard to an arbitrary Freud-type weight function. In current paper, one focuses on the weight functionto analyze its relative n and P n (z). Through above equation and orthogonality, we find that n (t) satisfy the first discrete Painlevé equation I Hierarchy and a high-order differential-difference equation, respectively. Then, we find that the asymptotic value of n is settled by Coulomb fluid. Additionally, we talk about P n (z) with = 0, including approximation for P n (0) and P ′ n (0), and bounds for P n (z) as n → ∞ are settled. KEYWORDS asymptotics, bounds, Freud-type weight, orthogonal polynomials, semiclassical MSC CLASSIFICATION 33C47; 34M55; 35C20; 65Q99 (z) = |z| e −z 6 +tz 2 ,where both z and t belonged to R, and > −1.Remark 1.1.(2) is a Freud-type weight, see Olver et al. 5Math Meth Appl Sci. 2020;43:5295-5313. wileyonlinelibrary.com/journal/mma

On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

“…However, the motive of the current work is different as it deals with the modified Freud-type weight function, which already involves a higher-order of the polynomial in the exponential factor in the semi-classical weight (4). A similar work for semi-classical Laguerre weight has been given in [11] and for the Freud-type weight in [18,36].…”

“…However, finding the moments explicitly has not been an easy task for some semi-classical weights. We refer to an interesting work in [15], which shows that the moments for the modified sextic and dodic weights are expressed in terms of generalized hypergoemetric functions and for quartic Freudian weights, see [12] (see also [18]).…”

“…Filipuk et al [23] found that the recurrence coefficients for the quartic Freudian weight |x| 2α+1 e −x 4 +tx 2 , x, t ∈ R, α > −1 are identified with solutions of the Painlevé IV and the first discrete Painlevé equation. Clarkson et al [12,15] gave a methodical investigation on Freud weights and some generalized work for [13] (see also [18,36]).…”

“…where K is a positive constant [5,10,16]. Certain properties of Freud-type polynomials can be found in [12,15,[17][18][19]. The authors in [12] considered monic orthogonal polynomials with respect to the quartic Freud-type weight function; that is, (m = 2 in Equation ( 2)) upon measure deformation by exp tx 2 .…”

Painlevé IV, $σ-$Form and the Deformed Hermite Unitary Ensembles

“…Remark 1. The equation (2.1) is called a lowering operator, see [2,3,9,14,[22][23][24]. With the help of (2.1) and (2.10), P n (z) also satisfies the raising operator equation…”