Zeros of quasi-orthogonal $q$-Laguerre polynomials

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.