We study the limiting behavior of the zeros of the Euler polynomials. When linearly scaled, they approach a definite curve in the complex plane related to the Szegö curve which governs the behavior of the roots of the Taylor polynomials associated to the exponential function. Further, under a conformal transformation, the scaled zeros are uniformly distributed.
ABSTRACT:For a subset S of positive integers let (n, S) be the set of partitions of n into summands that are elements of S. For every λ ∈ (n, S), let M n (λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on (n, S), and regard M n as a random variable. In this paper the limiting density of the (suitably normalized) random variable M n is determined for sets that are sufficiently regular. In particular, our results cover the case S = {Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman's coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes.
Let fi n be the expected order of a random permutation, that is, the arithmetic mean of the orders of the elements in the symmetric group S n . We prove that log/i n ~ c\/(n/\ogn) as n -> oo, where c = 2 / ( 2 I °° log log I --I dt\ 1. Overview
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