M. Foupouagnigni

Fourth order q -difference equation for the first associated of the q -classical orthogonal polynomials

Abstract We derive the fourth-order q-difference equation satisfied by the first associated of the q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials σ and τ which appear in the q-Pearson difference equation Dq(σ ϱ) = τϱ defining the weight ϱ of the q-classical orthogonal polynomials inside the q-Hahn tableau.

On difference equations for orthogonal polynomials on nonuniform lattices1

By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181–226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey–Wilson operator , following pioneering work by Magnus 1988, Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonal polynomials, Lecture Notes in Mathematics (Berlin: Springer), vol. 1329, pp. 261–278, connecting and divided-difference operators. Finally, we propose a method to look for polynomial solutions of linear divided-difference equations with polynomial coefficients.

Laguerre-Freud equations for the recurrence coefficients of D semi-classical orthogonal polynomials of class one

Abstract The Laguerre-Freud equations giving the recurrence coefficients β n , γ n of orthogonal polynomials with respect to a D ω semi-classical linear form are derived. D ω is the difference operator. The limit when ω → 0 are also investigated recovering known results. Applications to generalized Meixner polynomials of class one are also treated.

Fourth-order difference equation for the associated classical discrete orthogonal polynomials

We derive the fourth-order dierence equation satised by the associated order r of classical orthogonal polynomials of a discrete variable. The coecients of this equation are given in terms of the polynomials and which appear in the discrete Pearson equation ()= dening the weight (x) of the classical discrete orthogonal polynomials. c 1998 Elsevier Science B.V. All rights reserved. AMS classication: 33C25

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.

Characterization of the D w -Laguerre- Hahn Functionals

We give some characterization theorems for the D w -Laguerre-Hahn linear functionals and we extend the concept of the class of the usual Laguerre-Hahn functionals to the D w -Laguerre-Hahn functionals, recovering the classic results when w tends to zero. Moreover, we show that some transformations carried out on the D w -Laguerre-Hahn linear functionals lead to new D w -Laguerre-Hahn linear functionals. Finally, we analyze the class of the resulting functionals and we give some applications relative to the first associated Charlier, Meixner, Krawtchouk and Hahn orthogonal polynomials.

On Factorization and Solutions of q-difference Equations Satisfied by some Classes of Orthogonal Polynomials

We derive and factorize the fourth-order q-difference equations  satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations  and express the modified families in terms of the starting ones.

Difference equations for the co-recursive rth associated orthogonal polynomials of the Dq-Laguerre-Hahn class

We use some relations between the rth associated orthogonal polynomials of the Dq-Laguerre-Hahn class to derive the fourth-order q-difference equation satisfied by the co-recursive rth associated orthogonal polynomials of the Dq-Laguerre-Hahn class.When r=1 and for q-semi-classical situations, this q-difference equation factorizes as product of two second-order q-difference equations. Finally, we study some classical situations, and give some examples relative to the co-recursive associated discrete q-Hermite II orthogonal polynomials.

Fourth-order difference equation satisfied by the co-recursive of q -classical orthogonal polynomials

Abstract We derive the fourth-order q-difference equation satisfied by the co-recursive of q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials φ and ψ appearing in the q-Pearson difference equation D q (φρ)=ψρ defining the weight ρ of the q-classical orthogonal polynomials inside the q-Hahn tableau. Use of suitable change of variable and limit processes allow us to recover the results known for the co-recursive of the classical continuous and classical discrete orthogonal polynomials. Moreover, we describe particular situations for which the co-recursive of classical orthogonal polynomials are still classical and express these new families in terms of the starting ones.