Even Hypergeometric Polynomials and Finite Free Commutators

Jacob Campbell^a, Rafael Morales^b, Daniel Perales<sup>c

a</sup> Department of Mathematics, University of Virginia, VA, USA
^b Department of Mathematics, Baylor University, TX, USA
^c Department of Mathematics, University of Notre Dame, IN, USA

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.

Keywords: finite free probability, free commutator, genralized hypergeometric series, orthogonal polynomials.

MSC: 46L54, 33C20, 33C45

Received: April 15, 2025; in final form December 6, 2025; Published online December 21, 2025

Language: English

DOI: 10.3842/SIGMA.2025.108

ArXiv: 2502.00254