Quasi-orthogonality and zeros of some ${}_2φ_2$ and ${}_3φ_2$ polynomials (1912.00353v1)

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.