Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

Abstract

Abstract. We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials B n (x; λ) 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 E n (x; λ) via a simple relation linking them to the Apostol-Bernoulli polynomials.

Algebraic properties and Fourier expansions of two-dimensional Apostol–Bernoulli and Apostol–Euler polynomials

“…The results obtained generalize those in [2,6,8]. These are interesting results both from the theoretical and computational point of view, as they allow the use of Fourier approximation to both calculate and derive further relations between the various polynomial families involved.…”

“…These are interesting results both from the theoretical and computational point of view, as they allow the use of Fourier approximation to both calculate and derive further relations between the various polynomial families involved. For example [8] shows how the Fourier series gives an asymptotic approximation which is valid over compact complex domains.…”

General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials

“…It is worth mentioning that the Apostol-Bernoulli polynomials were firstly introduced by Apostol [6] (see also Srivastava [36] for a systematic study) in order to evaluate the value of the Hurwitz-Lerch zeta function. For some nice methods and results on these polynomials and numbers, one is referred to [7,8,26,32,33]. This paper is organized as follows.…”

Some formulas for the generalized Apostol-type polynomials and numbers

“…Since the publication of the above works by Luo and Srivastava, numerous properties for the generalized Apostol-Bernoulli polynomials, Euler and Genocchi polynomials have been studied. We refer to [5,7,8,15,17,20,21,25] for some related results on these Apostol-type polynomials and numbers. In [24,26], Ozden et al, constructed the following generating function:…”