Real roots of hypergeometric polynomials via finite free convolution

TL;DR

This paper investigates the real roots of hypergeometric polynomials using finite free convolutions, revealing their zero distribution and interlacing properties through free probability techniques.

Contribution

It introduces a novel approach to analyze hypergeometric polynomial roots via finite free convolutions and describes their asymptotic zero distribution using free probability laws.

Findings

Finite free convolutions preserve real zeros and interlacing.

Hypergeometric polynomial roots can be characterized as convolutions of elementary laws.

Asymptotic zero distribution expressed through free probability measures.

Abstract

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko-Pastur, reversed Marchenko-Pastur, and free beta laws, which has an independent interest within free probability.