Separating zeros of polynomials using an added interlacing point

TL;DR

This paper develops a unified framework to analyze and achieve interlacing of zeros between polynomial sequences by introducing an extra point, with applications to classical and non-orthogonal polynomials.

Contribution

It introduces a general method for ensuring interlacing of zeros using an added point, applicable to various polynomial families including orthogonal and non-orthogonal types.

Findings

Established conditions for interlacing of zeros with an added point.

Derived new interlacing results for Krawtchouk, Meixner, and Narayana polynomials.

Refined existing interlacing results for Jacobi and Laguerre polynomials.

Abstract

Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$ can be achieved by using a naturally arising extra point. Specifically, we analyse several general mixed recurrence relations that ensure the $n+1$ zeros of the polynomial $(x-E)\mathcal{P}_n(x)$ interlace with the $k$ zeros of $\mathcal{G}_k$, where $k=n$ or $n+1$. In addition, we show that imposing specific conditions on the extra point $E$ yields full interlacing between the zeros of $\mathcal{P}_n$ and $\mathcal{G}_k$ for a suitable choice of $n$. The approach provides a consolidated framework broadly applicable to both orthogonal and non-orthogonal polynomials and we illustrate this with new interlacing results for zeros of Krawtchouk, Meixner, and Narayana polynomials. We also illustrate that this general approach can be used to recover and refine existing results regarding the complete interlacing of zeros for classical Jacobi and Laguerre polynomials.