Interlacing of zeros of polynomials completed with two additional points

TL;DR

This paper studies how to complete the interlacing of zeros between polynomial pairs by adding two specific points, using recurrence relations and applying the results to classical orthogonal polynomials.

Contribution

It introduces a method to identify two extra points that achieve full interlacing of zeros in polynomial pairs, improving existing results and solving open questions.

Findings

Explicit extra points for Jacobi polynomial interlacing are provided.

New interlacing results are established for Pseudo-Jacobi polynomials.

Addresses an open problem for Meixner-Pollaczek polynomials with increased parameters.

Abstract

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra points required to achieve complete interlacing. We determine the precise positions of these two extra points relative to the zeros of the higher-degree polynomial, thereby establishing full interlacing results. The theory is applied to several classical families of orthogonal polynomials. In the Jacobi case, we improve earlier results by giving explicit extra points that complete the interlacing of $P_n^{(\alpha,\beta)}$ and $P_{n+1}^{(\alpha+1,\beta+1)}$. Second, we address an open question regarding the interlacing of zeros for Meixner-Pollaczek polynomials of consecutive degree with parameter increased by one. Finally, we establish new interlacing results for Pseudo-Jacobi polynomials.