On semiclassical orthogonal polynomials via polynomial mappings

An electrostatic interpretation of the zeros of sieved ultraspherical polynomials

“…Relations (4.1) and (4.2) establish a connection between sieved OP of the second kind and OPS obtained via a polynomial mapping as described in [3,Section 2]. This connection was established in a different way by Geronimo and Van Assche [9], and also in [17] (see also [6]).…”

“…Throughout this paper we will use the abbreviations OP and OPS for orthogonal polynomial(s) and orthogonal polynomial(s) sequence(s), respectively. In our first article [3] we obtained basic properties fulfilled by monic OPS {p n } n≥0 and {q n } n≥0 linked by a polynomial mapping, in the sense that there exist two polynomials π k and θ m , of (fixed) degrees k and m, respectively, where 0 ≤ m ≤ k − 1, such that p nk+m (x) = θ m (x) q n (π k (x)) , n = 0, 1, 2, . .…”

“…, under the assumption that one of the sequences {p n } n≥0 or {q n } n≥0 is a semiclassical OPS. In particular, we proved that if at least one of the sequences {p n } n≥0 or {q n } n≥0 is semiclassical then so is the other one, and we gave relations between their classes [3,Theorem 3.1].…”

“…Our present goal is to apply the results stated in [3] to the sieved OPS, introduced by Al-Salam, Allaway, and Askey [1], and subsequently studied by several authors (see e.g. [12,4,13,9,28,2,5,6,22,17]).…”

“…For reasons of economy of exposition, we assume familiarity with most of the results and notation appearing in Sections 2 and 3 of our previous article [3]. Let {p n } n≥0 be a monic OPS, so that, according to Favard's theorem it is characterized by a three-term recurrence such as…”