Large Degree Asymptotics of Generalized Bernoulli and Euler Polynomials

TL;DR

This paper derives comprehensive asymptotic expansions for large degree generalized Bernoulli and Euler polynomials, expanding on previous work limited to large parameters, using contour integrals and two-point Taylor expansions.

Contribution

It provides the first complete large degree asymptotic descriptions of these polynomials, including new expansion types and detailed analysis based on their generating functions.

Findings

Derived general asymptotic expansions for large n

Summarized known special case results

Introduced new expansion methods using two-point Taylor expansions

Abstract

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.