Even Hypergeometric Polynomials and Finite Free Commutators
Jacob Campbell, Rafael Morales, Daniel Perales
TL;DR
The paper develops a comprehensive framework to study even polynomials under finite free convolutions by translating them to positive-rooted polynomials via a degree-doubling map and its inverse. It introduces and analyzes even hypergeometric polynomials, their symmetrizations, and their behavior under additive, multiplicative, and generalized rectangular convolutions, providing numerous new examples from Laguerre, Hermite, and Jacobi families. The finite free commutator is explored through explicit constructions, with partial real-rootedness results and a wealth of examples linking to free probability. In the asymptotic regime, the authors connect these finite constructions to symmetric measures and the classical free probability transforms, establishing a robust bridge between finite and free theories and revealing new instances of free symmetrization and commutator behavior.
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.