Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials

TL;DR

This paper studies hypergeometric-Sobolev-type orthogonal polynomials, deriving their algebraic properties, recurrence relations, and zero behavior, especially how the mass parameter influences their zeros' locations.

Contribution

It provides explicit representations, recurrence relations, and zero analysis for Sobolev-type orthogonal polynomials associated with $q$-classical functionals, highlighting the impact of the mass parameter.

Findings

Explicit polynomial representations and recurrence relations.

Identification of parameter values affecting zero locations.

Analysis of zeros' behavior relative to the measure support.

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g \rangle _{S}:= \langle {\bf u}, f g\rangle +N (\mathscr D_q f)(\alpha) (\mathscr D _{q}g)(\alpha),\qquad \alpha\in \mathbb R, \quad N\ge 0, \] where $\bf u$ is a $q$-classical linear functional and $\mathscr D _{q}$ is the $q$-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear $q$-difference holonomic equation fulfilled by such polynomials. We present an analysis of the behaviour of its zeros function of the mass $N$. In particular, we in the exact values of $N$ such that the smallest (respectively, the greatest) zero of the studied polynomials is located outside of the support of the measure. We conclude this work considering two examples.