On the functional equation for classical orthogonal polynomials on lattices

Abstract

Necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials on lattices are stated. Moreover, the functional Rodrigues formula and a closed formula for the recurrence coefficients are presented.

A characterization of continuous $q$-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials

“…Since this work is an application of the theory developed in our previous paper [3], in the next sections, we suppose that the reader has [3] at hand and we shall use its notation, definitions, and results.…”

“…Multiplying both sides of this equality by π(X) and using (1.3) and (2.3), we obtain π(X)S q P n (X) = r [1] n P n+2 (X) + r [2] n P n+1 (X) + r [3] n P n (X) + r [4] n P n−1 (X) + r [5] n P n−2 (X)…”

Classical orthogonal polynomials revisited

“…We may now establish necessary and sufficient conditions ensuring the regularity of a given functional u ∈ P ′ satisfying (2.2). The next results was proved in [5, Theorem 2] (see also [3] in a more general context).…”

“…C4 ′ . 3 for each k ∈ N, there exist parameters r n such that (3.5) holds; C5. (Bochner) there exist polynomials φ and ψ and, for each n ≥ 0, a complex parameter λ n , with λ n = 0 if n ≥ 1, such that y = P n (x) is a solution of the second order ordinary differential equation φ(x)y ′′ + ψ(x)y ′ + λ n y = 0 , n ≥ 0 ;…”

On another characterization of Askey-Wilson polynomials

“…Theorem 4.1. [5] Let (P n ) n≥0 be a monic OPS with respect to u ∈ P * . Suppose that u satisfies the distributional equation D q (φu) = S q (ψu) , where φ(z) = az 2 + bz + c and ψ(z) = dz + e, with d = 0.…”