On semi‐classical d‐orthogonal polynomials

Abstract

In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation (ΦU)′=ΨU, where Φ and Ψ are d×d matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.

On Mittag–Leffler d-Orthogonal Polynomials

“…The latter expansion shows further that a linear combination of d-OPS could be again d-OPS [14]. It can be used also to construct semi-classical d-orthogonal polynomials examples [17] (of hypergeometric type).…”

Some Inverse Problems for d-Orthogonal Polynomials

“…In fact, {P n } is Hahn classical (semi-classical of order s=0) d-OPS, means that the sequence P [1] n = P ′ n+1 /(n + 1) is also d-orthogonal. In this case, the sequence {P n } itself is d-quasi-orthogonal of order two at most with respect to the vector linear form of P [1] n [66]. Consequently, we immediately have the following result Corollary 7.3.…”

“…In the usual orthogonality (d=1), it is well known that if a sequence of OPS {P n } is classical, then their derivative sequences of any order are again classical whereas this is not direct conclusion if d ≥ 2. In fact, the derivative sequence P [1] n is d-orthogonal with respect to V = ΦU [34,66]. Suppose that there exists matrix Υ such that ΥΦΨ = ΨΥΦ, then (Φ 1 V) ′ = Ψ 1 V with Φ 1 = ΥΦ and Ψ 1 = Φ ′ 1 + ΨΥ.…”