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

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 ${}_2{\varphi }_2$ and ${}_3{\varphi }_2$ 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.