Laguerre-Freud equations for Generalized Hahn polynomials of type I

Abstract

We derive a system of difference equations satisfied by the threeterm recurrence coefficients of some families of discrete orthogonal

Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI

“…Note that nonlinear recurrence relations for the recurrence coefficients of these orthogonal polynomials were found by Dominici in [8,Theorem 4], but these were of higher order and were not identified as discrete Painlevé equations. Our version (3.13)- (3.14) has the advantage that one can predict the asymptotic behavior of a 2 n and b n (or x n and y n ) as n → ∞ from them, and in Section 6 we conjectured this asymptotic behavior when the weights are on the lattice N and on the shifted lattice N + 1 − γ.…”

“…In Section 5 we make the connection with the σ-form of the sixth Painlevé equation (see Theorem 5.1). Note that Dominici already obtained non-linear recurrence relations for the recurrence coefficients in [8,Theorem 4] which he calls the Laguerre-Freud equations. These are however of higher order than two and neither they are identified as discrete Painlevé equations, nor is a connection made with Painlevé VI.…”

Pearson equations for discrete orthogonal polynomials: I. Generalized hypergeometric functions and Toda equations

“…In terms of the semi-infinite vector of monomials, (5) the moment matrix is written as 𝐺 = ⟨𝜌, 𝜒𝜒 ⊤ ⟩, and it becomes evident that this matrix is symmetric, that is, 𝐺 = 𝐺 ⊤ . The vector of monomials 𝜒 is an eigenvector of the shift matrix (6) that is, Λ𝜒 = 𝑥𝜒.…”

Pearson Equations for Discrete Orthogonal Polynomials: III. Christoffel and Geronimus transformations

“…We have also found that the contiguous relations fulfilled generalized hypergeometric functions determining the moments of the weight described for the squared norms of the orthogonal polynomials a discrete Toda hierarchy known as Nijho -Capel equation, see [44]. In [26] these ideas are applied to generalized Charlier, Meixner, and Hahn orthogonal polynomials extending the results of [22,47,27,28,29].…”

“…The mentioned relevance of discrete orthogonal polynomials it is also illustrated by numerous sections or chapters devoted to its discussion in excellent books on orthogonal polynomials such as [34,35,16,49]. For semiclassical discrete orthogonal polynomials the weight satisfies a discrete Pearson equation, we refer the reader to [24,25] and [22,23] and references therein for a comprehensive account. For the generalized Charlier and Meixner weights, Freud-Laguerre type equations for the coe cients of the three term recurrence has been discussed, see for example [20,27,28,29,47].…”