Cubic decomposition of a family of semiclassical orthogonal polynomials of class two

Several characterizations of third degree semiclassical linear forms of class two appearing via cubic decomposition

“…Recently, all semiclassical linear forms of class two such that their corresponding (MOPS) {W n } n≥0 satisfies W 3n (x) = P n (x 3 ) have been determined (see [29]). Only four canonical cases appear, up to an affine transformation.…”

“…Only four canonical cases appear, up to an affine transformation. Indeed, Theorem 1 [29] Let {W n } n≥0 be a (MOPS) fulfilling (2.10)-(2.12) with a 1 n = a 2 n = 0, n ≥ 0, and let w be its corresponding regular form. There exist only four families of semiclassical forms w of class two.…”

“…Later on, in [23] a symbolic approach is presented. We should point out that, problems concerning cubic decompositions in the framework of the semiclassical orthogonal polynomial sequences have not been considered in the literature up to the recent contributions [27], [28] and [29], for particular cases of semiclassical and H q -semiclassical orthogonal polynomials of class one and two.…”

“…Afterwards, we determine all the strict third degree linear forms which are semiclassical of class two arising via cubic decomposition. Indeed, among the semiclassical forms of class two arising via the (CD) W 3n (x) = P n (x 3 ), n ≥ 0, studied in [29], only three families, satisfying a certain condition, are of strict third degree. In Section 3, we state our main result, where a description and the identification of these linear forms are given.…”