Ladder operators and a differential equation for varying generalized Freud-type orthogonal polynomials

Abstract

In this paper, we introduce varying generalized Freud-type polynomials which are orthogonal with respect to a varying discrete Freud-type inner product. Our main goal is to give ladder operators for this family of polynomials as well as find a second-order differential–difference equation that these polynomials satisfy. To reach this objective, it is necessary to consider the standard Freud orthogonal polynomials and, in the meanwhile, we find new difference relations for the coefficients in the first-order differential equations that this standard family satisfies.

Differential, difference, and asymptotic relations for Pollaczek–Jacobi type orthogonal polynomials and their Hankel determinants

“…The ladder operator approach has been applied to solve many problems on orthogonal polynomials and Hankel determinants (see, e.g., Refs. 15,[23][24][25][26][27][28][29]. Following Chen and Its, 25 we have the lowering and raising operators for our Pollaczek-Jacobi type orthogonal polynomials:…”

Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials

“…The linearity follows directly from the definition. The product and quotient rules can be found, for example, in [2, f. (16)(17)].…”

“…Proof. From ( 7), (8), and applying the relationships (15)(16)(17)(18)(19), we deduce the following:…”

A differential equation for varying Krall-type orthogonal polynomials

“…. , m} and c i ̸ = c j if i ̸ = j. Polynomials orthogonal with respect to a varying inner product have been considered in different frameworks, for instance, in general contexts related to different types of weights (see, among others, [2,5,16,17,20,29,30,31,36,38]) or in Sobolev orthogonality (see, for instance, [1,18,34]).…”

“…In order to obtain this result, we follow the typical proof (see, for example, [3,8,9,13,20,24,26] and the references therein). From raising operator we have that…”