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-
Marcus, Polynomial convolutions and (finite) free probability, arXiv:2108.07054 [math.CO]
9 Pith papers cite this work. Polarity classification is still indexing.
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Proves Cosine and Hermite universality conjectures for roots of derivatives of even entire functions with real roots and for Jensen polynomials by establishing finite free probability limit theorems for repeated differentiation.
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.
Defines (n,d)-rectangular cumulants that linearize (n,d)-rectangular convolution in finite free probability and converge to q-rectangular free cumulants as d→∞ with 1+n/d→q.
Limiting root distribution after repeated fractional differentiation is the push-forward of the initial distribution under a characteristic flow of the log-potential PDE.
Introduces and studies the rectangular finite free heat flow as a dynamical system on polynomials with equivalent characterizations, root asymptotics, and connections to Calogero-Moser systems and mean curvature flow on Lie group orbits.
The finite R-transform of a polynomial differs from the Voiculescu R-transform of its empirical root distribution by O(N^{-1}), providing an analytic proof that finite free additive convolution converges to free additive convolution.
t-deformed convolution and cumulants on formal power series yield LLN and CLT analogues that recover classical convolution at t=-1 and finite free generators at t=d, with explicit infinitesimal generators for the associated semigroups.
Computational discovery via FlowBoost supports conjectures on the singular values of the coupling matrix E_n being 2^{-k/2} independent of n, a sharp p=2 critical exponent for p-Stam inequalities, and bifurcation of extremals for p<2.
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Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators
Limiting root distribution after repeated fractional differentiation is the push-forward of the initial distribution under a characteristic flow of the log-potential PDE.