A large family of semi-classical polynomials of class one

A family of second degree semi-classical forms of class s=1

“…It was shown in [5] that the corresponding MOPS {B n } n≥0 of u(β 0 ) satisfies (1.14) with β n = (−1) n β 0 , n ≥ 0 (see Lemma 2.2 in [5], p. 915). Then, we deduce the following result:…”

“…Only one case appears with Φ(x) = x(x 2 − 1). Let us keep the same notations of [5] where it was shown that these forms satisfy…”

“…In [5], the authors obtain all the semi-classical forms of class s = 1 satisfying (2.1) with β 0 = 1. Only one case appears with Φ(x) = x(x 2 − 1).…”

“…Later, Beghdadi [2] determined those of second degree. When β 0 = 0, Bouras and Alaya [5], gave all the semi-classical forms u(β 0 ) of class one. The aim of this work is to approach the problem of determining all the semiclassical forms u(β 0 ) of class s = 1 which are of the second degree when β 0 = 0.…”

“…Let u(β 0 ) be a regular form verifying the relation u(β 0 ) 2n+1 = β 0 u(β 0 ) 2n , n≥ 0, β 0 ∈ C, M. Sghaier ( ) Institut Supérieur d'Informatique de Medenine, Université de Gabès, Route Djerba-Km 3, 4119 Medenine, Tunisia e-mail: kkkbr@voila.fr which is equivalent to saying that its corresponding sequence of monic orthogonal polynomials {B n } n≥0 verifies the recurrence relation [5] B 0 (x) = 1,…”