Large Degree Asymptotics of Generalized Bernoulli and Euler Polynomials

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.