Fourier Series Expansion and Integral Representation of Apostol-Type Frobenius–Euler Polynomials of Complex Parameters and Order α

Abstract

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$ are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type Frobenius–Euler polynomials resulting in their expansions and integral representations.

1. Introduction

The Euler and Genocchi polynomials of order k are constructed by mixing the concepts of Euler and Genocchi numbers with the exponential polynomials. For $k=1$, we have the classical Euler and Genocchi polynomials, denoted by $E_n\left(x\right)$ and $G_n\left(x\right)$, respectively, defined by the following generating functions:

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{E}_n\left(x\right)\frac{t^n}{n!}& =\frac{2}{e^t+1}{e}^{xt},\left|t\right|<\pi ,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{G}_n\left(x\right)\frac{t^n}{n!}& =\frac{2t}{e^t+1}{e}^{xt},\left|t\right|<\pi .\hfill \end{array}$$

(2)

For $k>1$, we have the polynomials $E_n^{\left(k\right)}\left(x\right)$ and $G_n^{\left(k\right)}\left(x\right)$, which denote the Euler and Genocchi polynomials of order k, respectively, defined by

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{E}_n^{\left(k\right)}\left(x\right)\frac{t^n}{n!}& ={\left(\frac{2}{e^t+1}\right)}^k{e}^{xt},\left|t\right|<\pi ,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{G}_n^{\left(k\right)}\left(x\right)\frac{t^n}{n!}& ={\left(\frac{2t}{e^t+1}\right)}^k{e}^{xt},\left|t\right|<\pi .\hfill \end{array}$$

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In fact, $G_n\left(x\right)$ can be expressed in terms of $E_n\left(x\right)$ as follows:

$$G_{n+1}\left(x\right)=(n+1)E_n\left(x\right).$$

Similarly, $G_n^{\left(k\right)}\left(x\right)$ can be expressed in terms of $E_n^{\left(k\right)}\left(x\right)$ as follows:

$$G_{n+k}^{\left(k\right)}\left(x\right)={(n+k)}_k{E}_n^{\left(k\right)}\left(x\right).$$

These polynomials have a rich literature in the history of special functions. The two most recent studies on Genocchi polynomials are from the papers of Corcino-Corcino [1,2] on asymptotic approximations.

The Apostol-type Frobenius–Euler polynomials $H_n(x;u;\lambda )$ in the variable x are defined in (see [3,4,5]) by mixing Euler polynomials with the concept of Apostol and Frobenius polynomials as follows:

$$\sum _{n=0}^{\infty }{H}_n(x;u;\lambda )\frac{t^n}{n!}=\left(\frac{1-u}{\lambda e^t-u}\right)e^{xt},\left|t\right|<\left|log\left(\frac{u}{\lambda }\right)\right|,$$

(5)

where the logarithm function log is taken to be the principal branch. The Fourier series expansion of the Apostol-type Frobenius–Euler polynomials are given by

$$H_n(x;u;\lambda )=\frac{u-1}{u}n!{\left(\genfrac{}{}{0pt}{}{u}{\lambda }\right)}^x\sum _{k\in \mathbb{Z}}\frac{e^{2k\pi ix}}{{(2\pi ik-log\left(\frac{\lambda }{u}\right))}^{n+1}},$$

where $u,\lambda \in \mathbb{C}$ with $u\ne 1,\lambda \ne 1,u\ne \lambda$ and $0<x<1$ (see Theorem 1 of [3] (p. 5)), with $\mathbb{Z}$ and $\mathbb{C}$ denoting the sets of integers and complex numbers, respectively.

Recently, Alam et al. [6] defined a new class of Bell-based Apostol-type Frobenius–Euler polynomials and obtained some properties of these numbers and polynomials. Moreover, they derived summation formulas of Apostol-type Frobenius–Euler numbers and polynomials, expressed in terms of Apostol-type Bernoulli, Euler, and Genocchi polynomials. Furthermore, they proved several identities of Apostol-type Frobenius–Euler polynomials by using different analytical methods and generating function method. Lastly, they introduced parametric kinds of Apostol-type Frobenius–Euler polynomials and obtained some identities of these polynomials.

The Fourier series and integral representations for the Apostol–Euler polynomials and Apostol–Bernoulli polynomials were obtained by Luo (see [7]) using the Lipschitz summation formula, while in [8], using the Cauchy’s residue theorem in the complex plane, Bayad derived a Fourier series for the Apostol–Bernoulli, Apostol–Genocchi and Apostol–Euler polynomials. Recently, the Fourier series expansions for the Apostol-Genocchi, Apostol-Bernoulli, Apostol–Euler polynomials, Tangent Polynomials, Tangent–Bernoulli and Tangent–Genocchi polynomials of higher order were established in (see [9,10,11]).

In this paper, the Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$ will be introduced and their Fourier series expansions will also be derived. Moreover, their integral representations will be obtained by following the method of Luo [7]. The results in this study can be helpful in establishing the asymptotic approximation of Apostol-type Frobenius–Euler polynomials of order $\alpha$.

2. Methods

This study was facilitated with the use of residue theory—in particular, the Cauchy’s residue theorem [12] is applied to solve the contour integral representation of the polynomials. This theory demonstrates the technique of computing the symmetries of the defining equation of Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$.

3. Results and Discussions

The Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$ are defined by means of the generating function

$$\sum _{n=0}^{\infty }{H}_n^{\left(\alpha \right)}(x;u;\lambda )\frac{t^n}{n!}={\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }{e}^{xt},$$

(6)

where $u,\lambda \in \mathbb{C}$ with $u\ne 1,\lambda \ne 1$, $u\ne \lambda$ and $\alpha \in {\mathbb{Z}}^{+}$ (a set of positive integers). Now, let us consider first the function

$$f_n\left(t\right)={\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}.$$

The function $f_n\left(t\right)$ has a pole of order $n+1$ at $t=0$ and poles of order $\alpha$ at $t_k$ where $t_k$ are the zeroes of $\lambda e^t-u$. The values of $t_k$ are obtained as follows:

$$\begin{array}{cc}\hfill \lambda e^t-u& =0\hfill \\ \hfill \lambda e^t& =u\hfill \\ \hfill log\left(\lambda e^t\right)& =log\left(u\right)\hfill \\ \hfill log\left(\lambda \right)+log\left(e^t\right)& =log\left(u\right)\hfill \\ \hfill t+2k\pi i& =log\left(u\right)-log\left(\lambda \right)\hfill \\ \hfill t& =log\left(\frac{u}{\lambda }\right)-2k\pi i\hfill \\ \hfill t_k:=t& =2k\pi i-log\left(\frac{\lambda }{u}\right),fork\in \mathbb{Z}.\hfill \end{array}$$

Note that $log\left(\frac{\lambda }{u}\right)=\mathrm{Log}\left(\frac{\lambda }{u}\right)+2n\pi i$ where $\mathrm{Log}$ is the principal branch. We then integrate $f_n\left(t\right)$ around the circle with radius $(2N+ϵ)\pi$ where $ϵ\in \mathbb{R}$ (set of real numbers) such that $ϵ\pi i\pm log\left(\frac{\lambda }{u}\right)\ne 0\left(\mathrm{mod}2\pi i\right)$. This radius ensures that the circle $C_N$ does not pass any of the poles $t_k$. Then, from Cauchy’s residue theorem, we have

$${\int }_{C_N}{f}_n\left(t\right)dt=2\pi i\left(\mathrm{Res}(f_n\left(t\right),t=0)+\sum _{-N<k<N}\mathrm{Res}(f_n\left(t\right),t=t_k)\right).$$

(7)

The following lemma contains the limit of the integral in (7) as $N\to \infty$.

Lemma 1.

Let $\alpha \in {\mathbb{Z}}^{+}$, $\lambda ,u\in \mathbb{C}\setminus \{0,1\}$ with $\left|\lambda \right|\ne \left|u\right|$. For $0<x<\alpha ,n\ge 1$,

$$\underset{N\to \infty }{lim}{\int }_{C_N}{f}_n\left(t\right)dt=\underset{N\to \infty }{lim}{\int }_{C_N}{\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}dt=0,$$

where $C_N=\{t:|t|=(2N+ϵ)\pi \mathrm{and}ϵ\in \mathbb{R},(ϵ\pi i\pm log\left(\frac{\lambda }{u}\right)\ne 0\left(\mathrm{mod}2\pi i\right))\}$ and $C_N$ is positively-oriented.

Proof. 

Taking the modulus of the integral gives

$$\left|{\int }_{C_N}{\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}dt\right|\le \underset{t\in C_N}{sup}\left|{\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}\right|·2\pi \left[(2N+ϵ)\pi \right]$$

where $C_N=\{t:|t|=(2N+ϵ)\pi \mathrm{and}ϵ\in \mathbb{R},(ϵ\pi i\pm log\left(\frac{\lambda }{u}\right)\ne 0\left(\mathrm{mod}2\pi i\right))\}$.

We first show that $\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}$ is bounded above, for all $t\in C_N$.

Let $t=a+ib,a=\mathrm{Re}\left(t\right)$ and $b=\mathrm{Im}\left(t\right)$. Then,

$$|e^{xt}|=|e^{x(a+ib)}|=e^{xa}$$

and

$$\begin{array}{cc}\hfill \left|\frac{\lambda }{u}{e}^t-1\right|& =\left|\frac{\lambda }{u}{e}^{a+ib}-1\right|\hfill \\ \hfill & \ge \left|\left|\frac{\lambda }{u}{e}^{a+ib}\right|-\left|1\right|\right|\hfill \\ \hfill & =\left|\left|\frac{\lambda }{u}\right|e^a-1\right|\hfill \\ \hfill & =\sqrt{{\left(\left|\frac{\lambda }{u}\right|e^a-1\right)}^2}\hfill \\ \hfill & =e^a{\left({\left|\frac{\lambda }{u}\right|}^2-\frac{2\left|\frac{\lambda }{u}\right|}{e^a}+\frac{1}{e^{2a}}\right)}^{1/2}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}& \le \frac{e^{xa}}{e^{\alpha a}{\left({\left|\frac{\lambda }{u}\right|}^2-2\left|\frac{\lambda }{u}\right|e^{-a}+e^{-2a}\right)}^{\alpha /2}}\hfill \\ \hfill & =\frac{1}{e^{(\alpha -x)a}{\left({\left|\frac{\lambda }{u}\right|}^2-2\left|\frac{\lambda }{u}\right|e^{-a}+e^{-2a}\right)}^{\alpha /2}}.\hfill \end{array}$$

Let $D=e^{(\alpha -x)a}{\left({\left|\frac{\lambda }{u}\right|}^2-2\left|\frac{\lambda }{u}\right|e^{-a}+e^{-2a}\right)}^{\alpha /2}.$ Note that, with $0<x<\alpha$, $\alpha -x>0$. Now, we write $t=r{e}^{i\theta }=r(cos\theta +isin\theta )$ and see that $a=rcos\theta =\left[\right(2N+ϵ\left)\pi \right]cos\theta$, $-\pi <\theta \le \pi$. Thus, we have three cases:

Case 1. When $cos\theta >0$, $e^{(\alpha -x)a}\to \infty$ and $e^{-a},e^{-2a}\to 0$ as $N\to \infty$, which means that $D\to \infty$, and thus $\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}$ is bounded.

Case 2. When $cos\theta <0$, we write

$$D=e^{-xa}{\left({\left|\frac{\lambda }{u}\right|}^2{e}^{2a}-2\left|\frac{\lambda }{u}\right|e^a+1\right)}^{\alpha /2}$$

and see that $e^{-xa}\to \infty$ and $e^{2a},e^a\to 0$ as $N\to \infty$, which means that $D\to \infty$, and thus $\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}$ is bounded.

Case 3. When $cos\theta =0$, $e^{(\alpha -x)a}=1=e^{-a}=e^{-2a}$, which means that

$$D={\left({\left|\frac{\lambda }{u}\right|}^2-2\left|\frac{\lambda }{u}\right|+1\right)}^{\alpha /2}.$$

For $\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}$ to be bounded, ${\left|\frac{\lambda }{u}\right|}^2-2\left|\frac{\lambda }{u}\right|+1$ must not be zero. Note that ${\left|\frac{\lambda }{u}\right|}^2-2\left|\frac{\lambda }{u}\right|+1=0$, if and only if, $\left|\frac{\lambda }{u}\right|=1$. Hence, $\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}$ is bounded whenever $\left|\lambda \right|\ne \left|u\right|$.

Thus, in any case, $\forall t\in C_N$,

$$\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}\le K,$$

for some constant K provided that $\left|\lambda \right|\ne \left|u\right|$ when $\mathrm{Re}\left(t\right)=0$.

It then follows that, $\forall t\in C_N$,

$${\left|\frac{1-u}{u}\right|}^{\alpha }\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}\le K^{{}^{\prime }},$$

for some constant $K^{{}^{\prime }}$.

Finally, with $\left|t\right|=(2N+ϵ)\pi$,

$$\begin{array}{cc}\hfill \left|{\int }_{C_N}{\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}dt\right|& \le \left|{\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}\right|·2\pi \left[(2N+ϵ)\pi \right]\hfill \\ \hfill & ={\left|\frac{1-u}{u}\right|}^{\alpha }\frac{\left|e^{xt}\right|}{{\left|\frac{\lambda }{u}{e}^t-1\right|}^{\alpha }}\frac{2\pi \left[(2N+ϵ).\pi \right]}{|t^{n+1}|}\hfill \\ \hfill & \le \frac{2\pi K^{{}^{\prime }}}{{\left[(2N+ϵ)\pi \right]}^n},\hfill \end{array}$$

and clearly the last expression goes to 0 as $N\to \infty$ for $n\ge 1$. This proves the lemma. □

Using Lemma 1, Equation (7) becomes

$$\mathrm{Res}(f_n\left(t\right),t=0)=-\sum _{k\in \mathbb{Z}}\mathrm{Res}(f_n\left(t\right),t=t_k).$$

(8)

Fourier Series Expansion and Integral Representation

In this section, the main results of the paper will be discussed, which include the Fourier series expansion and integral representation of the Apostol-Frobenius–Euler polynomials of complex parameters and order $\alpha$. The following theorem contains the said Fourier series expansion.

Theorem 1.

Let $\alpha \in {\mathbb{Z}}^{+}$, $u,\lambda \in \mathbb{C}\setminus \{0,1\}$ with $\left|u\right|\ne \left|\lambda \right|$ and $0<x<\alpha$. We have

$$\begin{array}{cc}\hfill H_n^{\left(\alpha \right)}(x;u;\lambda )=n!{\left(\frac{u-1}{u}\right)}^{\alpha }{\left(\frac{u}{\lambda }\right)}^x\sum _{k\in \mathbb{Z}}\sum _{j=0}^{\alpha -1}& {(-1)}^j\left(\genfrac{}{}{0pt}{}{n+\alpha -j-1}{\alpha -j-1}\right)\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!}\hfill \\ \hfill & \times \frac{e^{2kx\pi i}}{{(2k\pi i-log\left(\frac{\lambda }{u}\right))}^{n+\alpha -j}},\hfill \end{array}$$

where $B_j^{\left(\alpha \right)}\left(x\right)$ denotes the Bernoulli polynomials of order α defined by (see [13])

$${\left(\frac{w}{e^w-1}\right)}^{\alpha }{e}^{xw}=\sum _{j=0}^{\infty }{B}_j^{\left(\alpha \right)}\left(x\right)\frac{w^j}{j!}.$$

Proof. 

We compute $\mathrm{Res}(f_n\left(t\right),t=0)$ and $\mathrm{Res}(f_n\left(t\right),t=t_k)$ as follows:

$$\begin{array}{cc}\hfill \mathrm{Res}(f_n\left(t\right),t=0)& =\frac{1}{n!}\underset{t\to 0}{lim}\frac{d^n}{d{t}^n}\left[{(t-0)}^{n+1}{f}_n\left(t\right)\right]\hfill \\ \hfill & =\frac{1}{n!}\underset{t\to 0}{lim}\frac{d^n}{d{t}^n}\left[t^{n+1}\left(\frac{1}{t^{n+1}}\sum _{m=0}^{\infty }{H}_m^{\left(\alpha \right)}(x;u;\lambda )\frac{t^m}{m!}\right)\right]\hfill \\ \hfill & =\frac{1}{n!}\underset{t\to 0}{lim}\frac{d^n}{d{t}^n}\left[\sum _{m=0}^{\infty }{H}_m^{\left(\alpha \right)}(x;u;\lambda )\frac{t^m}{m!}\right]\hfill \\ \hfill & =\frac{1}{n!}\underset{t\to 0}{lim}\sum _{m=n}^{\infty }{H}_m^{\left(\alpha \right)}(x;u;\lambda )\frac{t^{m-n}}{(m-n)!}\hfill \\ \hfill & =\frac{H_n^{\left(\alpha \right)}(x;u;\lambda )}{n!}\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill \mathrm{Res}(f_n\left(t\right),t=t_k)& =\frac{1}{(\alpha -1)!}\underset{t\to t_k}{lim}\frac{d^{\alpha -1}}{d{t}^{\alpha -1}}\left[{(t-t_k)}^{\alpha }{f}_n\left(t\right)\right]\hfill \\ \hfill & =\frac{1}{(\alpha -1)!}\underset{t\to t_k}{lim}\frac{d^{\alpha -1}}{d{t}^{\alpha -1}}\left[{(t-t_k)}^{\alpha }{\left(\frac{1-u}{\lambda e^t-u}\right)}^{\alpha }\frac{e^{xt}}{t^{n+1}}\right]\hfill \\ \hfill & =\frac{1}{(\alpha -1)!}\underset{t\to t_k}{lim}\frac{d^{\alpha -1}}{d{t}^{\alpha -1}}\left[{(t-t_k)}^{\alpha }\frac{{(1-u)}^{\alpha }}{u^{\alpha }{\left(\frac{\lambda }{u}{e}^t-1\right)}^{\alpha }}\frac{e^{xt}}{t^{n+1}}\right]\hfill \\ \hfill & =\frac{1}{(\alpha -1)!}{\left(\frac{1-u}{u}\right)}^{\alpha }\underset{t\to t_k}{lim}\frac{d^{\alpha -1}}{d{t}^{\alpha -1}}\left[\frac{{(t-t_k)}^{\alpha }}{{\left(\frac{\lambda }{u}{e}^t-1\right)}^{\alpha }}\frac{e^{xt}}{t^{n+1}}\right].\hfill \end{array}$$

(10)

Since $e^{-t_k}=exp(-2k\pi i+log\left(\frac{\lambda }{u}\right))=\frac{\lambda }{u}$, we can write ${\left(\frac{\lambda }{u}{e}^t-1\right)}^{\alpha }={\left(e^{-t_k}{e}^t-1\right)}^{\alpha }={\left(e^{(t-t_k)}-1\right)}^{\alpha }$. Thus, we have

$$\begin{array}{cc}\hfill \frac{{(t-t_k)}^{\alpha }}{{\left(\frac{\lambda }{u}{e}^t-1\right)}^{\alpha }}\frac{e^{xt}}{t^{n+1}}& =\frac{{(t-t_k)}^{\alpha }}{{\left(e^{(t-t_k)}-1\right)}^{\alpha }}\frac{e^{xt}}{t^{n+1}}\hfill \\ \hfill & =\left(\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^m}{m!}\right)\frac{e^{xt}}{t^{n+1}}\hfill \\ \hfill & =\left(e^{xt}\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^m}{m!}\right)t^{-(n+1)},\hfill \end{array}$$

where $B_m^{\left(\alpha \right)}$ denotes the Bernoulli numbers of order $\alpha$ defined by (see [13])

$${\left(\frac{w}{e^w-1}\right)}^{\alpha }=\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{w^m}{m!}.$$

Applying the Leibniz Rule on differentiation,

$$\begin{array}{cc}\hfill \frac{d^{\alpha -1}}{d{t}^{\alpha -1}}& \left[\frac{{(t-t_k)}^{\alpha }}{{\left(\frac{\lambda }{u}{e}^t-1\right)}^{\alpha }}\frac{e^{xt}}{t^{n+1}}\right]\hfill \\ \hfill & =\frac{d^{\alpha -1}}{d{t}^{\alpha -1}}\left[\left(e^{xt}\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^m}{m!}\right)t^{-(n+1)}\right]\hfill \\ \hfill & =\sum _{j=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\alpha -1}{j}\right)\frac{d^j}{d{t}^j}\left(e^{xt}\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^m}{m!}\right)·\frac{d^{\alpha -1-j}}{d{t}^{\alpha -1-j}}{t}^{-(n+1)},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \frac{d^j}{d{t}^j}& \left(e^{xt}\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^m}{m!}\right)\hfill \\ \hfill & =\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)\frac{d^l}{d{t}^l}\sum _{m=0}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^m}{m!}·\frac{d^{j-l}}{d{t}^{j-l}}{e}^{xt}\hfill \\ \hfill & =\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)\sum _{m=l}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^{m-l}}{(m-l)!}·x^{j-l}{e}^{xt}\hfill \\ \hfill & =e^{xt}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)x^{j-l}\sum _{m=l}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^{m-l}}{(m-l)!}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \frac{d^{\alpha -1-j}}{d{t}^{\alpha -1-j}}& t^{-(n+1)}\hfill \\ \hfill & =\frac{d^{\alpha -1-j}}{d{t}^{\alpha -1-j}}{t}^{-n-1}\hfill \\ \hfill & =(-n-1)(-n-2)...(-n-\alpha +1+j)t^{-n-\alpha +j}\hfill \\ \hfill & ={(-1)}^{\alpha -1-j}(n+\alpha -j-1)...(n+2)(n+1)t^{-(n+\alpha -j)}\hfill \\ \hfill & ={(-1)}^{\alpha -1-j}\frac{(n+\alpha -j-1)!}{n!}{t}^{-(n+\alpha -j)}.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \frac{d^{\alpha -1}}{d{t}^{\alpha -1}}& \left[\frac{{(t-t_k)}^{\alpha }}{{\left(\frac{\lambda }{u}{e}^t-1\right)}^{\alpha }}\frac{e^{xt}}{t^{n+1}}\right]\hfill \\ \hfill & =\sum _{j=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\alpha -1}{j}\right){(-1)}^{\alpha -1-j}\frac{(n+\alpha -j-1)!}{n!}{t}^{-(n+\alpha -j)}\hfill \\ \hfill & \times e^{xt}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)x^{j-l}\sum _{m=l}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^{m-l}}{(m-l)!}.\hfill \end{array}$$

(11)

Applying (11) to (10) yields

$$\begin{array}{cc}\hfill & \mathrm{Res}(f_n\left(t\right),t=t_k)\hfill \\ \hfill & =\frac{1}{(\alpha -1)!}{\left(\frac{1-u}{u}\right)}^{\alpha }\underset{t\to t_k}{lim}\sum _{j=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\alpha -1}{j}\right){(-1)}^{\alpha -1-j}\frac{(n+\alpha -j-1)!}{n!}{t}^{-(n+\alpha -j)}\hfill \\ \hfill & \times \underset{t\to t_k}{lim}{e}^{xt}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)x^{j-l}\sum _{m=l}^{\infty }{B}_m^{\left(\alpha \right)}\frac{{(t-t_k)}^{m-l}}{(m-l)!}.\hfill \end{array}$$

Observe that $B_m^{\left(\alpha \right)}\frac{{(t-t_k)}^{m-l}}{(m-l)!}\to 0$ as $t\to t_k$ except when $m=l$. Then,

$$\begin{array}{cc}\hfill & \mathrm{Res}(f_n\left(t\right),t=t_k)\hfill \\ \hfill & =\frac{1}{(\alpha -1)!}{\left(\frac{1-u}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}\left(\genfrac{}{}{0pt}{}{\alpha -1}{j}\right){(-1)}^{\alpha -1-j}\frac{(n+\alpha -j-1)!}{n!}{t}_k^{-(n+\alpha -j)}\hfill \\ \hfill & \times e^{x{t}_k}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)x^{j-l}{B}_l^{\left(\alpha \right)}\hfill \\ \hfill & =\frac{{(-1)}^{\alpha }}{(\alpha -1)!}{\left(\frac{1-u}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}\frac{(\alpha -1)!}{j!(\alpha -j-1)!}{(-1)}^{-1-j}\frac{(n+\alpha -j-1)!}{n!}{t}_k^{-(n+\alpha -j)}\hfill \\ \hfill & \times e^{x{t}_k}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)x^{j-l}{B}_l^{\left(\alpha \right)}\hfill \\ \hfill & ={\left(\frac{u-1}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}{(-1)}^{j+1}\frac{(n+\alpha -j-1)!}{n!(\alpha -j-1)!}\frac{e^{x{t}_k}}{j!t_k^{n+\alpha -j}}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)x^{j-l}{B}_l^{\left(\alpha \right)}.\hfill \end{array}$$

Recall from [13] that $B_j^{\left(\alpha \right)}\left(x\right)={\sum }_{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right)B_l^{\left(\alpha \right)}{x}^{j-l}$. Thus,

$$\begin{array}{cc}\hfill \mathrm{Res}(f_n\left(t\right),t=t_k)& ={\left(\frac{u-1}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}{(-1)}^{j+1}\left(\genfrac{}{}{0pt}{}{n+\alpha -j-1}{\alpha -j-1}\right)\frac{e^{x{t}_k}}{j!t_k^{n+\alpha -j}}{B}_j^{\left(\alpha \right)}\left(x\right)\hfill \\ \hfill & ={\left(\frac{u-1}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}{(-1)}^{j+1}\left(\genfrac{}{}{0pt}{}{n+\alpha -j-1}{\alpha -j-1}\right)\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!}\frac{e^{x{t}_k}}{t_k^{n+\alpha -j}}.\hfill \end{array}$$

Substituting $t_k=2k\pi i-log\left(\frac{\lambda }{u}\right)$, we find

$$\begin{array}{cc}\hfill \mathrm{Res}(f_n\left(t\right),t=t_k)& ={\left(\frac{u-1}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}{(-1)}^{j+1}\left(\genfrac{}{}{0pt}{}{n+\alpha -j-1}{\alpha -j-1}\right)\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!}\hfill \\ \hfill & \times \frac{exp\left(2kx\pi i+log{\left(\frac{u}{\lambda }\right)}^x\right)}{{(2k\pi i-log\left(\frac{\lambda }{u}\right))}^{n+\alpha -j}}\hfill \\ \hfill & ={\left(\frac{u-1}{u}\right)}^{\alpha }{\left(\frac{u}{\lambda }\right)}^x\sum _{j=0}^{\alpha -1}{(-1)}^{j+1}\left(\genfrac{}{}{0pt}{}{n+\alpha -j-1}{\alpha -j-1}\right)\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!}\hfill \\ \hfill & \times \frac{e^{2kx\pi i}}{{(2k\pi i-log\left(\frac{\lambda }{u}\right))}^{n+\alpha -j}}.\hfill \end{array}$$

(12)

Combining the residues (9) and (12) with Equation (8) yields

$$\begin{array}{cc}\hfill H_n^{\left(\alpha \right)}(x;u;\lambda )=n!{\left(\frac{u-1}{u}\right)}^{\alpha }{\left(\frac{u}{\lambda }\right)}^x\sum _{k\in \mathbb{Z}}\sum _{j=0}^{\alpha -1}& {(-1)}^j\left(\genfrac{}{}{0pt}{}{n+\alpha -j-1}{\alpha -j-1}\right)\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!}\hfill \\ \hfill & \times \frac{e^{2kx\pi i}}{{(2k\pi i-log\left(\frac{\lambda }{u}\right))}^{n+\alpha -j}}.\hfill \end{array}$$

This completes the proof. □

Now, let us consider an integral representation of the Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$.

Theorem 2.

For $n\in \mathbb{N}$ (set of natural numbers), $0<x<\alpha$, $\left|\xi \right|<\frac{1}{2},\xi \in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill & H_n^{\left(\alpha \right)}(x;u;-u{e}^{2\xi \pi i})\hfill \\ \hfill & ={\left(\frac{u-1}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}{(-1)}^j\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!(\alpha -j-1)!}{e}^{-2\xi x\pi i}\times \hfill \\ \hfill & \times {\int }_0^{\infty }\frac{M(n;x,v)cosh\left(2\xi \pi v\right)+iN(n;x,v)sinh\left(2\xi \pi v\right)}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv,\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\hfill & M(n;x,v)=\left[e^{\pi v}cos\left(\pi x-\frac{(n+\alpha -j)\pi }{2}\right)-e^{-\pi v}cos\left(\pi x+\frac{(n+\alpha -j)\pi }{2}\right)\right],\hfill \\ \hfill & N(n;x,v)=\left[e^{\pi v}sin\left(\pi x-\frac{(n+\alpha -j)\pi }{2}\right)+e^{-\pi v}sin\left(\pi x+\frac{(n+\alpha -j)\pi }{2}\right)\right].\hfill \end{array}$$

Proof. 

From (1), we have

$$\begin{array}{cc}\hfill H_n^{\left(\alpha \right)}(x;u;\lambda )=& {\left(\frac{u-1}{u}\right)}^{\alpha }\sum _{j=0}^{\alpha -1}{(-1)}^j\frac{B_j^{\left(\alpha \right)}\left(x\right)}{j!(\alpha -j-1)!}\hfill \\ \hfill & \times {\left(\frac{u}{\lambda }\right)}^x(n+\alpha -j-1)!\sum _{k\in \mathbb{Z}}\frac{e^{2kx\pi i}}{{(2k\pi i-log\left(\frac{\lambda }{u}\right))}^{n+\alpha -j}}.\hfill \end{array}$$

(14)

Now, we look at

$$\mathsf{\Theta }={\left(\frac{u}{\lambda }\right)}^x(n+\alpha -j-1)!\sum _{k\in \mathbb{Z}}\frac{e^{2kx\pi i}}{{(2k\pi i-log\left(\frac{\lambda }{u}\right))}^{n+\alpha -j}}.$$

(15)

Taking $\lambda =-u{e}^{2\xi \pi i}$, $k\mapsto -k$, we have

$$\begin{array}{cc}\hfill \mathsf{\Theta }& ={\left(\frac{1}{e^{\pi i}{e}^{2\xi \pi i}}\right)}^x(n+\alpha -j-1)!\sum _{k\in \mathbb{Z}}\frac{e^{-2kx\pi i}}{{(-2k\pi i-log\left(e^{\pi i}{e}^{2\xi \pi i}\right))}^{n+\alpha -j}}\hfill \\ \hfill & =e^{-(2\xi +1)x\pi i}(n+\alpha -j-1)!\sum _{k\in \mathbb{Z}}\frac{e^{-2kx\pi i}}{{(-2k\pi i-2\xi \pi i-\pi i)}^{n+\alpha -j}}\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}(n+\alpha -j-1)!}{{(-\pi i)}^{n+\alpha -j}}\sum _{k\in \mathbb{Z}}\frac{e^{-2kx\pi i}}{{(2k+2\xi +1)}^{n+\alpha -j}}\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}(n+\alpha -j-1)!}{{(-\pi i)}^{n+\alpha -j}}\left(\sum _{k=0}^{\infty }\frac{e^{-2kx\pi i}}{{(2k+2\xi +1)}^{n+\alpha -j}}\right.\hfill \\ \hfill & +\left.\sum _{k=1}^{\infty }\frac{e^{2kx\pi i}}{{(-2k+2\xi +1)}^{n+\alpha -j}}\right)\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}(n+\alpha -j-1)!}{{(-\pi i)}^{n+\alpha -j}}\left(\sum _{k=0}^{\infty }\frac{e^{-2kx\pi i}}{{(2k+2\xi +1)}^{n+\alpha -j}}\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}\sum _{k=1}^{\infty }\frac{e^{2kx\pi i}}{{(2k-2\xi -1)}^{n+\alpha -j}}\right)\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left(\sum _{k=0}^{\infty }{e}^{-2kx\pi i}\frac{(n+\alpha -j-1)!}{{(2k+2\xi +1)}^{n+\alpha -j}}\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}\sum _{k=1}^{\infty }{e}^{2kx\pi i}\frac{(n+\alpha -j-1)!}{{(2k-2\xi -1)}^{n+\alpha -j}}\right).\hfill \end{array}$$

(16)

Here,

$$\begin{array}{cc}\hfill (2k+2\xi +1)>0& if\left|\xi \right|<\frac{1}{2},\xi \in \mathbb{R}andk\ge 0,\hfill \\ \hfill (2k-2\xi -1)>0& if\left|\xi \right|<\frac{1}{2},\xi \in \mathbb{R}andk\ge 1.\hfill \end{array}$$

Applying the integral formula (see [7] (p. 294, Equation (3.2)))

$${\int }_0^{\infty }{t}^n{e}^{-at}dt=\frac{n!}{a^{n+1}}(n=0,1,...;\Re \left(a\right)>0)$$

in (16), we find that

$$\begin{array}{cc}\hfill \mathsf{\Theta }& =\frac{e^{-(2\xi +1)x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left(\sum _{k=0}^{\infty }{e}^{-2kx\pi i}{\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{-(2k+2\xi +1)t}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}\sum _{k=1}^{\infty }{e}^{2kx\pi i}{\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{-(2k-2\xi -1)t}dt\right)\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{-(2\xi +1)t}\sum _{k=0}^{\infty }{e}^{-2(x\pi i+t)k}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{(2\xi +1)t}\sum _{k=1}^{\infty }{e}^{2(x\pi i-t)k}dt\right)\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{-(2\xi +1)t}\frac{1}{1-e^{-2(x\pi i+t)}}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{(2\xi +1)t}\frac{e^{2(x\pi i-t)}}{1-e^{2(x\pi i-t)}}dt\right)\hfill \\ \hfill & =\frac{e^{-(2\xi +1)x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{-(2\xi +1)t}\frac{e^{2t}}{e^{2t}-e^{-2x\pi i}}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }{t}^{n+\alpha -j-1}{e}^{(2\xi +1)t}\frac{e^{2x\pi i}}{e^{2t}-e^{2x\pi i}}dt\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }\frac{e^{-x\pi i}}{e^{2t}-e^{-2x\pi i}}{e}^{(1-2\xi )t}{t}^{n+\alpha -j-1}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }\frac{e^{x\pi i}}{e^{2t}-e^{2x\pi i}}{e}^{(1+2\xi )t}{t}^{n+\alpha -j-1}dt\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }\frac{1}{e^{2t+x\pi i}-e^{-x\pi i}}·\frac{2(e^{-2x\pi i}-e^{-2t})e^{x\pi i}}{2(e^{-x\pi i}-e^{-2t+x\pi i})}{e}^{(1-2\xi )t}{t}^{n+\alpha -j-1}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }\frac{1}{e^{2t-x\pi i}-e^{x\pi i}}·\frac{2(e^{2x\pi i}-e^{-2t})e^{-x\pi i}}{2(e^{x\pi i}-e^{-2t-x\pi i})}{e}^{(1+2\xi )t}{t}^{n+\alpha -j-1}dt\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{2{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }\frac{2(e^{-2x\pi i}-e^{-2t})e^{x\pi i}}{(e^{2t}+e^{-2t})-(e^{2x\pi i}+e^{-2x\pi i})}{e}^{(1-2\xi )t}{t}^{n+\alpha -j-1}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }\frac{2(e^{2x\pi i}-e^{-2t})e^{-x\pi i}}{(e^{2t}+e^{-2t})-(e^{2x\pi i}+e^{-2x\pi i})}{e}^{(1+2\xi )t}{t}^{n+\alpha -j-1}dt\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{2{(-\pi i)}^{n+\alpha -j}}\left({\int }_0^{\infty }\frac{(e^{-2x\pi i}-e^{-2t})e^{x\pi i}}{cosh2t-cos2x\pi }{e}^{(1-2\xi )t}{t}^{n+\alpha -j-1}dt\right.\hfill \\ \hfill & \left.+{(-1)}^{n+\alpha -j}{\int }_0^{\infty }\frac{(e^{2x\pi i}-e^{-2t})e^{-x\pi i}}{cosh2t-cos2x\pi }{e}^{(1+2\xi )t}{t}^{n+\alpha -j-1}dt\right).\hfill \end{array}$$

Making the substitution $t=\pi v$ and noting that ${(-1/i)}^{n+\alpha -j}=e^{(n+\alpha -j)\pi i/2}$ and ${(-1)}^{n+\alpha -j}=e^{-(n+\alpha -j)\pi i}$, we find

$$\begin{array}{cc}\hfill & \mathsf{\Theta }=\frac{e^{-2\xi x\pi i}}{2·{\pi }^{n+\alpha -j}}\left({\int }_0^{\infty }\frac{e^{(n+\alpha -j)\pi i/2}(e^{-2x\pi i}-e^{-2\pi v})e^{x\pi i}}{cosh2\pi vs.-cos2x\pi }{e}^{(1-2\xi )\pi v}{\left(\pi v\right)}^{n+\alpha -j-1}\pi dv\right.\hfill \\ \hfill & \left.+{\int }_0^{\infty }\frac{e^{-(n+\alpha -j)\pi i/2}(e^{2x\pi i}-e^{-2\pi v})e^{-x\pi i}}{cosh2\pi vs.-cos2x\pi }{e}^{(1+2\xi )\pi v}{\left(\pi v\right)}^{n+\alpha -j-1}\pi dv\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{2}\left({\int }_0^{\infty }\frac{e^{i\left[-\pi x+(n+\alpha -j)\pi /2\right]}-e^{i\left[\pi x+(n+\alpha -j)\pi /2\right]}{e}^{-2\pi v}}{cosh2\pi vs.-cos2x\pi }{e}^{(1-2\xi )\pi v}{v}^{n+\alpha -j-1}dv\right.\hfill \\ \hfill & \left.+{\int }_0^{\infty }\frac{e^{i\left[\pi x-(n+\alpha -j)\pi /2\right]}-e^{i\left[-\pi x-(n+\alpha -j)\pi /2\right]}{e}^{-2\pi v}}{cosh2\pi vs.-cos2x\pi }{e}^{(1+2\xi )\pi v}{v}^{n+\alpha -j-1}dv\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{2}\left({\int }_0^{\infty }\frac{e^{\pi v}{e}^{i\left[-\pi x+(n+\alpha -j)\pi /2\right]}{e}^{-2\xi \pi v}-e^{-\pi v}{e}^{i\left[\pi x+(n+\alpha -j)\pi /2\right]}{e}^{-2\xi \pi v}}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right.\hfill \\ \hfill & \left.+{\int }_0^{\infty }\frac{e^{\pi v}{e}^{i\left[\pi x-(n+\alpha -j)\pi /2\right]}{e}^{2\xi \pi v}-e^{-\pi v}{e}^{i\left[-\pi x-(n+\alpha -j)\pi /2\right]}{e}^{2\xi \pi v}}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right).\hfill \end{array}$$

Let $A=[\pi x-(n+\alpha -j)\pi /2]$ and $B=[\pi x+(n+\alpha -j)\pi /2]$. Then,

$$\begin{array}{cc}\hfill \mathsf{\Theta }& =\frac{e^{-2\xi x\pi i}}{2}\left({\int }_0^{\infty }\frac{e^{\pi v}{e}^{i(-A)}{e}^{-2\xi \pi v}-e^{-\pi v}{e}^{iB}{e}^{-2\xi \pi v}}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right.\hfill \\ \hfill & \left.+{\int }_0^{\infty }\frac{e^{\pi v}{e}^{iA}{e}^{2\xi \pi v}-e^{-\pi v}{e}^{i(-B)}{e}^{2\xi \pi v}}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right)\hfill \\ \hfill & =\frac{e^{-2\xi x\pi i}}{2}\left({\int }_0^{\infty }\frac{\left(e^{\pi v}cosA-e^{-\pi v}cosB\right)\left(e^{2\xi \pi v}+e^{-2\xi \pi v}\right)}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right.\hfill \\ \hfill & \left.+{\int }_0^{\infty }i\frac{\left(e^{\pi v}sinA+e^{-\pi v}sinB\right)\left(e^{2\xi \pi v}-e^{-2\xi \pi v}\right)}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right)\hfill \\ \hfill & =e^{-2\xi x\pi i}\left({\int }_0^{\infty }\frac{\left(e^{\pi v}cosA-e^{-\pi v}cosB\right)cosh\left(2\xi \pi v\right)}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right.\hfill \\ \hfill & \left.+{\int }_0^{\infty }i\frac{\left(e^{\pi v}sinA+e^{-\pi v}sinB\right)sinh\left(2\xi \pi v\right)}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv\right)\hfill \\ \hfill & =e^{-2\xi x\pi i}{\int }_0^{\infty }\frac{M(n;x,v)cosh\left(2\xi \pi v\right)+iN(n;x,v)sinh\left(2\xi \pi v\right)}{cosh2\pi vs.-cos2x\pi }{v}^{n+\alpha -j-1}dv,\hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\hfill M(n;x,v)& =\left[e^{\pi v}cos\left(\pi x-\frac{(n+\alpha -j)\pi }{2}\right)-e^{-\pi v}cos\left(\pi x+\frac{(n+\alpha -j)\pi }{2}\right)\right],\hfill \\ \hfill N(n;x,v)& =\left[e^{\pi v}sin\left(\pi x-\frac{(n+\alpha -j)\pi }{2}\right)+e^{-\pi v}sin\left(\pi x+\frac{(n+\alpha -j)\pi }{2}\right)\right].\hfill \end{array}$$

Replacing (15) with (17) in (14), we obtain the desired integral representation. □

4. Conclusions and Recommendations

The Fourier series expansion of the Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$ was derived using the Cauchy residue theorem. Consequently, an integral representation of the Apostol-type Frobenius–Euler polynomials of complex parameters and order $\alpha$ was obtained using the Fourier series expansion of the polynomials. We recommend establishing the asymptotic approximation of these Apostol-type Frobenius–Euler polynomials.