For the Riemann xi-function, the Jensen polynomials J^{d,n} are hyperbolic whenever n^3 log^2(n+2) ≥ K d^5, and their scaled zeros converge to Wigner's semicircle law in this joint limit.
Real roots of hypergeometric polynomials via finite free convolution
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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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A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function
For the Riemann xi-function, the Jensen polynomials J^{d,n} are hyperbolic whenever n^3 log^2(n+2) ≥ K d^5, and their scaled zeros converge to Wigner's semicircle law in this joint limit.