Classification of Laguerre–Hahn orthogonal polynomials of class one

Abstract

We study orthogonal polynomials related to Stieltjes functions satisfying Riccati type differential equations with polynomial coefficients,We derive recurrences for the three-term recurrence relation coefficients of the orthogonal polynomials, including connections with some forms of discrete Painlevé equations. K E Y W O R D SLaguerre-Freud equations, orthogonal polynomials, Painlevé equations, Stieltjes function M S C ( 2 0 1 0 ) 33C45, 33C47, 42C05 MOTIVATIONOrthogonal polynomials { ( ) = + lower degree terms} ≥0 on the real line may be fully characterized by the orthogonality relation1) or, equivalently, by a three-term recurrence relation [27] +1 ( ) = ( − ) ( ) − −1 ( ), = 1, 2, … , (1.2) with 0 ( ) = 1, 1 ( ) = − 0 and ≠ 0, ≥ 1. Here, , is the Kronecker delta, and is a positive Borel measure supported on a finite or infinite interval of the real line, , with finite moments 244

Laguerre–Hahn orthogonal polynomials on the real line

“…Taking into account the results from Theorem 2, the lemma above can be used to find a representation for the matrices Y n related to AS = BS 2 + CS + D. Note that the representation (37) depends on the solution of two matrix linear systems, (35) and (36), which are not always trivially solved. Let us start with some remarks on the solution of (36).…”

“…Lemma 2. Let S be a Stieltjes function that satisfies the Riccati differential equation AS = BS 2 + CS + D. Let {P n } n≥0 be the corresponding SMOP such that the equations (25), AY n = B n Y n −Y n C, hold, with the corresponding system (36). Let the following assumption hold:…”

“…4]). To AS = BS 2 + CS + D associate AY n = B n Y n − Y n C and the corresponding systems (35) and (36), AL = CL, AP n = B n P n . Let G be a domain in the complex plane such that the entries of the matrices B n /A and C/A are integrable in G. Assume assumption (40).…”

“…It remains to study the representation (37) for Laguerre-Hahn orthogonal polynomials by solving the two general systems (35) and (36).…”

“…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).…”