Any free additive infinitely divisible distribution is the weak limit of root distributions of Appell polynomials f_n(∂_z)z^n for Laguerre-Pólya sequences f_n, with extensions to multiplicative cases, rectangular convolution, and limiting Cauchy distribution for Jensen polynomials of the Riemann Xi-
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Finite free perpetuities are defined as degree-n monic polynomials solving a truncated perpetuity equation; the paper proves existence, uniqueness, real nonnegative zeros for admissible (A,B), and weak convergence of root distributions to free perpetuity laws.
Geometric polynomials have new asymptotic behaviors in the complex plane and on (-1,0), specific consecutive zero distances in the bulk, and orthogonality relations that apply verbatim to Eulerian polynomials.
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P\'olya--Schur problems and free probability
Any free additive infinitely divisible distribution is the weak limit of root distributions of Appell polynomials f_n(∂_z)z^n for Laguerre-Pólya sequences f_n, with extensions to multiplicative cases, rectangular convolution, and limiting Cauchy distribution for Jensen polynomials of the Riemann Xi-
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Finite free perpetuities
Finite free perpetuities are defined as degree-n monic polynomials solving a truncated perpetuity equation; the paper proves existence, uniqueness, real nonnegative zeros for admissible (A,B), and weak convergence of root distributions to free perpetuity laws.
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Asymptotics and zero behaviour of geometric polynomials
Geometric polynomials have new asymptotic behaviors in the complex plane and on (-1,0), specific consecutive zero distances in the bulk, and orthogonality relations that apply verbatim to Eulerian polynomials.