Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials

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.


Introduction
The Fourier series of a periodic function can be written exponentially as (see [9,p. 19,Eq. (2.2)]) f (x) = ∞ n=-∞ a n e inwx ; w = 2π T , the coefficients a n and a n are computed by a n = 1 T 2π w 0 e -inwt f (t) dt and a n = 1 T Here a n is the complex conjugate of a n . The Frobenius-Euler polynomials and the Frobenius-Euler numbers play an important role in the number of theories and classical analysis. In particular, 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 (see [1, 4-6, 10, 13, 19]). The Frobenius-Euler polynomials H n (x; u) in the variable x are defined by means of the generating function (see [11, p. 268]) 1u e zu e xz = ∞ n=0 H n (x; u) z n n! , |z| < log(u) , when x = 0, H n (u) denotes the so-called Frobenius-Euler numbers. H n (x; -1) = E n (x) denotes the Euler polynomials (see [16,17]). The Fourier series representation of the Frobenius-Euler polynomials are given by (see [2,p. 8,Corollary 4]) if u, ∈ C with u = 1, and 0 < x < 1. The Frobenius-Genocchi polynomials G F n (x; u) in the variable x are defined by the generating function (see [2, p. 3 when x = 0, G F n (u) denotes the so-called Frobenius-Genocchi numbers, then the Fourier series representation of (3) is given by It is well known that Apostol-type Frobenius-Euler polynomials H n (x; u; λ) in the variable x are defined by means of the generating function (see [4, p. 164 The Fourier series representation of the Apostol-type Frobenius-Euler polynomials is given by (see [2,p. 5,Theorem 1]) Also, the Fourier series representation of Apostol-type Frobenius-Genocchi polynomials is given by (see [2,p. 13,Theorem 11]) which makes sense if u, λ ∈ C with u = 1, λ = 1, u = λ and 0 < x < 1. For parameters λ, u ∈ C, u = λ and a, b, c ∈ R + whit a = b, of generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating functions (see [18, p. 9, Definition 4.1]): if x = 0 in (9) then we get H n (a, b, c; u; λ), which denotes the generalized Apostol-type Frobenius-Euler numbers (see [18, p. 9]). For n = 0 and a, b ∈ R + , whit a = b, u, λ ∈ C with u = λ it is then true that H n (a, b; u; λ) = 1-u λ-u . For n > 0 we have (see [18, p. 10, Theorem 4.2]) λ ln b + H(a, b; u; λ) n -uH n (a, b; u; λ) = (ln a) n .
As an example, the generalized Apostol-type Frobenius-Euler numbers and polynomials are (with the help of MAPLE) as follows.
The generalized Apostol-type Frobenius-Euler numbers: The generalized Apostol-type Frobenius-Euler polynomials: These polynomials are commonly said to be of Euler type, and they have been studied by various authors in different applications of practical importance (see [1,12,21]). On the other hand, the Hurwitz-Lerch zeta function (z, s, a) is defined as (see [15, p. 296 and e(s) > 1 for every |z |= 1.
For z = 1 in (10) we have the Hurwitz zeta functions Recently, there was defined a new family of Lerch-type zeta function, interpolating a certain class of higher-order Apostol-type numbers and Apostol-type polynomials (cf. [20]). We will use (10) and (11) in Theorems 5.1 and 5.2.

Fourier expansion of generalized Apostol-type Frobenius-Euler polynomials
In this section, we get the Fourier expansions for the generalized Apostol-type Frobenius-Euler polynomials.
This is the Fourier expansion for Frobenius-Euler polynomials obtained in (see [2, p. 8,

Integral representation of the generalized Apostol-type Frobenius-Euler polynomials
In this section, we will show the integral representation of generalized Apostol-type Frobenius-Euler polynomials.
The result obtained in Corollary 4.2 is a new integral representation for the Apostol-Euler polynomials H n (x; 1, e, e; -1; e 2π iξ ) = E n (x; e 2π iξ ).
Next, we obtain the integral representation of Apostol-type Frobenius-Genocchi polynomials.
Using ( 1 -i ) n = e nπ i 2 and (-1) n = e -nπ i , making the substitution t = πv and simplifying, we complete the proof.
Proof Returning to (17), setting u = e 2π iξ , k − → -k and using the well-known integral formula (5), we complete the proof.
Proof Returning to (4) and setting u = e 2π iξ , k − → -k and using the well-known integral formula (5), we complete the proof.

Conclusions
In this article, we showed the Fourier series representation of generalized Apostol-type Frobenius-Euler polynomials by using the proof of the Cauchy residue theorem. The result presented generalizes several Fourier series representations for polynomial families known to date. Also, we proved an integral representation for this and other known polynomial families. Finally, we presented the explicit formula in rational arguments in terms of the Zeta Hurwit Lerch and Zeta Hurwit functions for the generalized Apostol-type Frobenius Euler polynomials also said to be of Euler type.