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.

12 pages