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.
The purpose of this note is to characterize those orthogonal polynomials sequences (Pn) n≥0 for whichwhere Dq is the Askey-Wilson operator, π is a polynomial of degree at most 2, and (an) n≥0 , (bn) n≥0 and (cn) n≥0 are sequences of complex numbers such that cn = 0 for n = 1, 2, . . . . This gives an additional input to a conjecture posed by Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable,
In [Castillo & Mbouna, Indag. Math. 31 (2020) [223][224][225][226][227][228][229][230][231][232][233][234], the concept of π N -coherent pairs of order (m, k) with index M is introduced. This definition, implicitly related with the standard derivative operator, automatically leaves out the so-called discrete orthogonal polynomials. The purpose of this note is twofold: first we use the (discrete) Hahn difference operator and rewrite the known results in this framework; second, as an application, we describe exhaustively the (discrete) self-coherent pairs in the situation whether M = 0, N ≤ 2, and (m, k) = (1, 0). This is proved by describing in a unified way the classical orthogonal polynomials with respect to Jackson's operator as special or limiting cases of a four parametric family of q−polynomials.
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