Bounds for extreme zeros of Meixner–Pollaczek polynomials

Abstract

In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function $c_{2k}\left(x\right),k\in N_0$, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner–Pollaczek polynomial. When $p_n$ is orthogonal with respect to a weight $w\left(x\right)$ and $g_{n-m}$ is orthogonal with respect to the weight $c_{2k}\left(x\right)w\left(x\right)$, we show that $k\in \{0,1,\dots ,m\}$ is a necessary and sufficient condition for existence of a connection formula involving a polynomial $G_{m-1}$ of degree $(m-1)$, such that the $(n-1)$ zeros of $G_{m-1}{g}_{n-m}$ and the $n$ zeros of $p_n$ interlace. We analyse the new inner bounds for the extreme zeros of Meixner–Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials.