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Ap\'ery Polynomials and the multivariate Saddle Point Method
The Ap\'ery polynomials and in particular their asymptotic behavior play an
essential role in the understanding of the irrationality of \zeta(3). In this
paper, we present a method to study the asymptotic behavior of the sequence of
the Ap\'ery polynomials ((B_{n})_{n=1}^{\infty}) in the whole complex plane as
(n\rightarrow \infty). The proofs are based on a multivariate version of the
complex saddle point method. Moreover, the asymptotic zero distributions for
the polynomials ((B_{n})_{n=1}^{\infty}) and for some transformed Ap\'ery
polynomials are derived by means of the theory of logarithmic potentials with
external fields, establishing a characterization as the unique solution of a
weighted equilibrium problem. The method applied is a general one, so that the
treatment can serve as a model for the study of objects related to the Ap\'ery
polynomials.Comment: 19 page
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