Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 108, 33 pages      arXiv:2502.00254      https://doi.org/10.3842/SIGMA.2025.108

Even Hypergeometric Polynomials and Finite Free Commutators

Jacob Campbell a, Rafael Morales b and Daniel Perales c
a) 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

Received April 15, 2025, in final form December 06, 2025; Published online December 21, 2025

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.

Key words: finite free probability; free commutator; genralized hypergeometric series; orthogonal polynomials.

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