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.
Let F n (x) be the partition polynomial ∑ n k=1 p k (n)x k where p k (n) is the number of partitions of n with k parts. We emphasize the computational experiments using degrees up to 70, 000 to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of F n (x) have two scales of orders n and √ n and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.