Fourier expansion and integral representation generalized Apostol-type Frobenius\u2013Euler polynomials

Alejandro Urieles,Daniel Bedoya,William Ramírez,María José Ortega

doi:10.1186/s13662-020-02988-0

Abstract

The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius–Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz–Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius–Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius–Genocchi polynomials.

Highlights

IntroductionThe Fourier series of a periodic function can be written exponentially as (see [9, p. 19, Eq (2.2)])
The Fourier series of a periodic function can be written exponentially as (see [9, p. 19, Eq (2.2)]) ∞ f (x) = aneinwx; n=–∞2π w=, T the coefficients an and an are computed by 1 an = T2π w e–inwtf (t) dt and2π w einwtf (t) dt.Here an is the complex conjugate of an
The Frobenius–Euler polynomials appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials

Summary

Introduction

The Fourier series of a periodic function can be written exponentially as (see [9, p. 19, Eq (2.2)]). We obtained the Fourier expansion of generalized Apostol-type Frobenius– Euler polynomials and its integral representation to show the explicit formula at rational arguments for these polynomials in terms of the Hurwitz zeta function. 3 are revealed the Fourier expansions for the generalized Apostol-type Frobenius–Euler polynomials, and several corollaries for other families of known polynomials. 4, we obtain the integral representation of generalized Apostol-type Frobenius–Euler polynomials, that is, Theorem 4.1. 5 secures the explicit formula at rational arguments in terms of Hurwitz zeta function of generalized Apostol-type Frobenius–Euler polynomials, that is, Theorem 5.1. It is well known that Apostol-type Frobenius–Euler polynomials Hn(x; u; λ) in the variable x are defined by means of the generating function The Fourier series representation of the Apostol-type Frobenius–Euler polynomials is given by We will use (10) and (11) in Theorems 5.1 and 5.2

Fourier expansion of generalized Apostol-type Frobenius–Euler polynomials

Conclusions