Interlacing of zeros from different sequences of Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials

TL;DR

This paper investigates the interlacing properties of zeros between different polynomial sequences, specifically Meixner-Pollaczek, Pseudo-Jacobi, and Continuous Hahn polynomials, using a new mixed recurrence relation.

Contribution

It introduces a novel mixed recurrence condition that ensures interlacing of zeros between different polynomial sequences and applies it to specific orthogonal polynomials.

Findings

Established interlacing conditions for zeros of different polynomial sequences.

Derived new interlacing results for Meixner-Pollaczek, Pseudo-Jacobi, and Continuous Hahn polynomials.

Provided a framework for analyzing zeros interlacing via mixed recurrence relations.

Abstract

In this paper we consider interlacing of the zeros of polynomials from different sequences $\{p_n\}$ and $\{g_n\}$. In our main result we consider a mixed recurrence equation necessary for existence of a linear term $(x-A)$ so that the $(n+1)$ zeros of $(x-A)g_n(x)$ interlace with the $n$ zeros of $p_n$. We apply our result to Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials to obtain new interlacing results for the zeros of polynomials of the same degree from different polynomial sequences.