On the Krall-type discrete polynomials

Abstract

In this paper we present a unified theory for studying the so-called Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of q-Krall polynomials are considered in detail.

Limit relations between q-Krall type orthogonal polynomials

“…In this section, we will introduce the q-Krall-type orthogonal polynomials. In a very recent paper [9] the authors introduce the "discrete" Krall polynomials as a perturbation of a classical or semiclassical discrete linear functional and they develop a general theory in order to find some algebraic properties such as TTRR, SODE, etc. In this paper we focus our attention on the special case when the starting functional C is a q-classical functional [21].…”

“…where C is the linear functional (1) and x 0 , x 1 ∈ R. In [9] a general theory for solving this problem (when N mass points are added) has been presented, nevertheless only two examples were considered in details.…”

“…The explicit expression of the polynomials P A,B n (s) q orthogonal with respect to the linear functional U (16) is given by [9] (it is assumed that the polynomials P n and P n have the same leading coefficient)…”

“…In a recent paper [9] the case when C is a discrete semiclassical or q-semiclassical linear functional was considered in details. Here we will focus our attention on the case when C is a q-classical linear functional and we will construct the Krall-type polynomials associated with the q-classical families of the so-called q-Hahn tableau [8,19].…”

“…Before continuing let us point out that the above families satisfy a three-term recurrence relation and a second order linear difference equation. For more details see [9].…”