On zeros of quasi-orthogonal Meixner polynomials

TL;DR

This paper studies the zeros of quasi-orthogonal Meixner polynomials, proving conjectures about their zero bounds, interlacing properties, and non-orthogonality for certain parameter ranges.

Contribution

It proves a conjecture on the lower bound of the first positive zero and establishes bounds for zeros of quasi-orthogonal Meixner polynomials, also showing non-orthogonality in specific cases.

Findings

Proved the conjecture on the lower bound of the first positive zero.

Identified bounds for the first few zeros of quasi-orthogonal Meixner polynomials.

Showed that certain Meixner polynomial sequences cannot be orthogonal with respect to any positive measure.

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 $n\in\mathbb{N}_{\geqq 3}.$