Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

TL;DR

This paper provides detailed asymptotic analysis of Apostol-Bernoulli and Apostol-Euler polynomials, including explicit estimates and oscillatory behavior, using Fourier series methods applicable in complex analysis.

Contribution

It introduces explicit asymptotic estimates for Apostol-Bernoulli polynomials and extends these results to Apostol-Euler polynomials through a simple relation, advancing understanding of their behavior.

Findings

Explicit asymptotic estimates for Apostol-Bernoulli polynomials.

Analysis of oscillatory phenomena in certain cases.

Transfer of results to Apostol-Euler polynomials via a linking relation.

Abstract

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $\mathcal{B}_{n}(x;\lambda)$ in detail. The starting point is their Fourier series on $[0,1]$ which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials $\mathcal{E}_{n}(x;\lambda)$ via a simple relation linking them to the Apostol-Bernoulli polynomials.