Quasi-orthogonality and zeros of some ${}_2\phi_2$ and ${}_3\phi_2$ polynomials

TL;DR

This paper extends known results to establish the quasi-orthogonality and zero distribution properties of certain ${}_2 heta_2$ and ${}_3 heta_2$ polynomials, using $q$-orthogonal polynomials and contiguous relations.

Contribution

It provides a $q$-extension of a previous result and analyzes the zeros and interlacing properties of specific quasi-orthogonal $q$-polynomials.

Findings

Established quasi-orthogonality of certain ${}_2 heta_2$ and ${}_3 heta_2$ polynomials.

Analyzed the zero distribution and interlacing properties of these polynomials.

Extended the understanding of zeros of $q$-Laguerre polynomials in specific parameter ranges.

Abstract

We state and prove the $q$-extension of a result due to Johnston and Jordaan (cf. \cite{Johnston-2015}) and make use of this result, the orthogonality of $q$-Laguerre, little $q$-Jacobi, $q$-Meixner and Al-Salam-Carlitz I polynomials as well as contiguous relations satisfied by the polynomials, to establish the quasi-orthogonality of certain $_2\phi_2$ and $_3\phi_2$ polynomials. The location and interlacing properties of the real zeros of these quasi-orthogonal polynomials are studied. Interlacing properties of the zeros of $q$-Laguerre quasi-orthogonal polynomials $L_n^{(\delta)}(z;q)$ when $-2<\delta<-1$ with those of $L_{n-1}^{(\delta+1)}(z;q)$ and $L_n^{(\delta+1)}(z;q)$ are also considered.