Real roots of hypergeometric polynomials via finite free convolution

2023/09/19 by Andrei Martı́nez-Finkelshtein, Martinez-Finkelshtein, Andrei, R. Morales +3 · 4 citations
Mathematics · #33C20 #33C45 #42C05 #46L54 #Advanced Mathematical Identities #Benford’s Law and Fraud Detection #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2309.10970

openalex publication_date 2023/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Zeros and exponential profiles of polynomials II: Examples
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• Generalized Meixner-type free gamma distributions:convolution formulas and potential correspondence

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