On zeros of quasi-orthogonal Meixner polynomials

Abstract

For each fixed value of $\beta$ in the range $-2<\beta <-1$ and $0<c<1$, we investigate interlacing properties of the zeros of polynomials of consecutive degree for $M_n(x;\beta ,c)$ and $M_k(x,\beta +t,c)$, $k\in \{n-1,n,n+1\}$ and $t\in \{0,1,2\}$. We prove the conjecture in [K. Driver and A. Jooste, Quasi-orthogonal Meixner polynomials, Quaest. Math. 40 (4) (2017), 477-490] on a lower bound for the first positive zero of the quasi-orthogonal order $1$ polynomial $M_n(x;\beta +1,c)$ and identify upper and lower bounds for the first few zeros of quasi-orthogonal order $2$ Meixner polynomials $M_n(x;\beta ,c)$. We show that a sequence of Meixner polynomials $\{M_n(x;\beta ,c){\}}_{n=3}^{\infty }$ with $-2<\beta <-1$ and $0<c<1$ cannot be orthogonal with respect to any positive measure by proving that the zeros of $M_{n-1}(x;\beta ,c)$ and $M_n(x;\beta ,c)$ do not interlace for any $n\in {\mathbb{N}}_{\geqq 3}.$