Zeros of quasi-orthogonal $q$-Laguerre polynomials

TL;DR

This paper studies the zero distribution and interlacing properties of quasi-orthogonal $q$-Laguerre polynomials within a specific parameter range, providing new bounds for their smallest zeros.

Contribution

It introduces new results on zero interlacing and bounds for the least zero of quasi-orthogonal $q$-Laguerre polynomials, extending understanding of their zero behavior.

Findings

Zeros of quasi-orthogonal $q$-Laguerre polynomials are interlaced with those of related orthogonal polynomials.

New bounds for the smallest zero of these polynomials are established.

Interlacing properties depend on polynomial degree and parameter shifts.

Abstract

We investigate the interlacing of zeros of polynomials of different degrees within the sequences of $q$-Laguerre polynomials $\left\{\tilde{L}_n^{(\delta)}(z;q)\right\}_{n=0}^{\infty}$ characterized by $\delta\in(-2,-1).$ The interlacing of zeros of quasi-orthogonal polynomials $\tilde{L}_n^{(\delta)}(z;q)$ with those of the orthogonal polynomials $\tilde{L}_m^{(\delta+t)}(z;q), m,n\in\mathbb{N}, t\in\{1,2\}$ is also considered. New bounds for the least zero of the (order $1$) quasi-orthogonal $q$-Laguerre polynomials are derived.