On the modifications of semi-classical orthogonal polynomials on nonuniform lattices

Discrete semi-classical orthogonal polynomials of class one on quadratic lattices

“…Semi-classical orthogonal polynomials on quadratic lattices may be defined through: (i) a Pearson equation for the linear functional [9,10],…”

“…When m = 0 we get the so-called classical polynomials [9,18]. The class of a linear functional L on quadratic lattices was defined in [10], as the non-negative integer given by…”

“…(95)The lattice x(s) and the polynomials p, r that follow from (7) arex(s) = s(s + γ + δ + 1) , p(x) = x + 1 4 , r(x) = x + (γ + δ + 1) 2 4 . (96){P n } n≥0 is related to a linear functional L that satisfies D(φL) = M(ψL), where the polynomials φ, ψ are given by[10] φ(x) = (−1 + 2N + δ − γ)x + N (1 + γ)(1 + γ + δ) , ψ(x) = −2x + 2N (1 + γ) . (97) The Stieltjes function satisfies(18), A DS = C MS + D, with A, C given by(20), thus,A = Mφ + 2r(x) − 1 Mψ , C = −1 − Dφ + Mψ .…”

Symmetric Semi-classical Orthogonal Polynomials of Class One on q-Quadratic Lattices

“…We take the difference equation satisfied by the Stieltjes functions, say ADS = CMS + D, where A, C, D are polynomials, subject to restrictions deg(A) ≤ 3, deg(C) ≤ 2. According to the classification from [8], this is the so-called class one (see Section 2.2). Our main goal is to give a closed form expression for the recurrence coefficients of orthogonal polynomials in the symmetric case, that is, when one of the recurrence coefficients is zero (cf.…”

“…where A, C, D are irreducible polynomials (in x). In general, the polynomials A, C, D in (14) satisfy, in the account of (1), (8), and 12,…”