The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their Extensions

Abstract

We study symmetric sequences of orthogonal polynomials of class two related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients. We show difference equations for the recurrence coefficients of the orthogonal polynomials as well as for related quantities. Some of such recurrences are identified with discrete Painlevé equations.

Laguerre–Hahn orthogonal polynomials on the real line

“…, N , where F j are non-linear functions in the β n 's and γ n 's. In the Laguerre-Hahn setting, some of these recurrences have been identified as forms of discrete Painlevé equations in [39,36,37]. The derivation of discrete Painlevé equations for the recurrence coefficient of orthogonal polynomials is a well-know topic in the literature of special functions (see, for instance, [53,54,85] and the recent monograph [84]), more information on this topic will be given in Subsection 4.2.2.…”

“…Three canonical cases for the polynomial A appear: A(x) = Recently, these classes have been revisited in [36,37]: in [36], it is given a recursive way to compute β n and γ n in Laguerre-Hahn class s = 1,…”

“…withf n ,ĝ n explicit expressions given in terms of the coefficients of the Riccati equation; in [37], on the symmetric class s = 2, it is given a fourth order difference equation for γ n [37,Cor. 4],…”

“…Unlike the semi-classical case, the Laguerre-Hahn setting falls short on such a study. To the best of the author's knowledge, the derivation of discrete Painlevé equations for the recurrence coefficients of Laguerre-Hahn orthogonal polynomials on the real line is limited to [39,36,37], with focus on the classes s = 1 and s = 2. The derivation of the referred discrete Painlevé equations is purely algebraic, it is based on the use of the equations enclosed in (30), with the identities (32) and 33playing a fundamental role.…”

“…Remark . Relations (32) and (33) play a fundamental role in [36,37,39], to derive discrete Painlevé equations for the recurrence coefficients of Laguerre-Hahn orthogonal polynomials (see more details in Subsection 4.2.2).…”