Large degree asymptotics of generalized Bernoulli and Euler polynomials

摘要

Asymptotic expansions are given for large values of n of the generalized Bernoulli polynomials B_n~μ (z) and Euler polynomials E_n~μ (z). In a previous paper López and Temme (1999) these polynomials have been considered for large values of μ, with n fixed. In the literature no complete description of the large n asymptotics of the considered polynomials is available. We give the general expansions, summarize known results of special cases and give more details about these results. We use two-point Taylor expansions for obtaining new type of expansions. The analysis is based on contour integrals that follow from the generating functions of the polynomials.