Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large values of $\mu$, 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.

asymptotic expansions, generalized Bernoulli polynomials, generalized Euler polynomials
Bernoulli and Euler numbers and polynomials (msc 11B68)
Academic Press
Journal of Mathematical Analysis and Applications

López, J.L, & Temme, N.M. (2010). Large degree asymptotics of generalized Bernoulli and Euler polynomials. Journal of Mathematical Analysis and Applications, 363(1), 197–208.