Even Hypergeometric Polynomials and Finite Free Commutators

TL;DR

This paper explores even polynomials, especially hypergeometric types, within finite free probability, providing new examples and linking their root distributions to symmetric measures in free probability.

Contribution

It introduces new classes of even hypergeometric polynomials and finite free commutators, expanding understanding of their root behaviors and connections to free probability.

Findings

New examples of even hypergeometric polynomials are provided.

Relationships between root distributions and symmetric measures are established.

Finite free convolution properties of these polynomials are characterized.

Abstract

We study in detail the class of even polynomials and their behavior with respect to finite free convolutions. To this end, we use some specific hypergeometric polynomials and a variation of the rectangular finite free convolution to understand even real-rooted polynomials in terms of positive-rooted polynomials. Then, we study some classes of even polynomials that are of interest in finite free probability, such as even hypergeometric polynomials, symmetrizations, and finite free commutators. Specifically, we provide many new examples of these objects, involving classical families of special polynomials (such as Laguerre, Hermite, and Jacobi). Finally, we relate the limiting root distributions of sequences of even polynomials with the corresponding symmetric measures that arise in free probability.