# Higher-order Euler-type polynomials and their applications

## Abstract

In this paper, we construct generating functions for higher-order Euler-type polynomials and numbers. By using the generating functions, we obtain functional equations related to a generalized partial Hecke operator and Euler-type polynomials and numbers. A special case of higher-order Euler-type polynomials is eigenfunctions for the generalized partial Hecke operators. Moreover, we give not only some properties, but also applications for these polynomials and numbers.

AMS Subject Classification:08A40, 11F25, 11F60, 11B68, 30D05.

## 1 Introduction

In this section, we define generalized partial Hecke operators and we give some notation for these operators. Also, we define generalized Euler-type polynomials, Apostol-Bernoulli polynomials and Frobenius-Euler polynomials.

Throughout this paper, we use the following notations:

$\mathbb{N}=\left\{1,2,\dots \right\}$, ${\mathbb{N}}_{0}=\left\{0,1,2,\dots \right\}=\mathbb{N}\cup \left\{0\right\}$. Also, as usual, denotes the set of integers, denotes the set of real numbers and denotes the set of complex numbers. We assume that $ln\left(z\right)$ denotes the principal branch of the multi-valued function $ln\left(z\right)$ with an imaginary part $\mathrm{\Im }\left(ln\left(z\right)\right)$ constrained by $-\pi <\mathrm{\Im }\left(ln\left(z\right)\right)\le \pi$. Furthermore, ${0}^{n}=1$ if $n=0$, and ${0}^{n}=0$ if $n\in \mathbb{N}$.

$N\left(M\right)=\left({N}_{1},{N}_{2},\dots ,{N}_{M}\right),$

where $M\in \mathbb{N}$ and ${N}_{1},{N}_{2},\dots ,{N}_{M}\in \mathbb{N}$.

Let $a\in \mathbb{N}$ and ${\chi }_{a,N\left(M\right)}$ be a function depending on $a,{N}_{1},{N}_{2},\dots ,{N}_{M}$ such that

${\chi }_{a,N\left(M\right)}:{\mathbb{N}}_{0}\to \mathbb{C}.$

${\chi }_{a,N\left(M\right)}$ is defined by

${\chi }_{a,N\left(M\right)}\left(k\right)=\prod _{j=1}^{M}{\xi }^{k}\left({N}_{j}\right),$

where $0\le k\le a-1$, $j\in \left\{1,2,\dots ,M\right\}$ and

$\xi \left({N}_{j}\right)={e}^{\frac{2\pi i}{{N}_{j}}}.$

${\chi }_{a,N\left(M\right)}$ satisfies the following properties:

1. (i)

${\chi }_{a,N\left(M\right)}$ is a periodic function with ${N}_{1}{N}_{2}\cdots {N}_{M}$.

2. (ii)

If we take ${N}_{1}\ge 2$ and ${N}_{2}={N}_{3}=\cdots ={N}_{M}=1$, we have

${\chi }_{a,\left({N}_{1},1,1,\dots ,1\right)}\left(k\right)={\xi }^{k}\left({N}_{1}\right){\xi }^{k}\left(1\right){\xi }^{k}\left(1\right)\cdots {\xi }^{k}\left(1\right)={\xi }^{k}\left({N}_{1}\right).$

We note that replacing $N\left(M\right)$ by $\left({N}_{1},1,1,\dots ,1\right)$, ${\chi }_{a,N\left(M\right)}$ is reduced to ${\xi }^{k}\left({N}_{1}\right)$ (cf. ).

Let $\mathbb{C}\left[x\right]$ be a ring of polynomials with complex coefficients. By using ${\chi }_{a,N\left(M\right)}$, we give the following definition.

Definition 1.1 

Let $P\in \mathbb{C}\left[x\right]$. The generalized partial Hecke operator of ${T}_{{\chi }_{a,N\left(M\right)}}$ is defined by

${T}_{{\chi }_{a,N\left(M\right)}}\left(P\left(x\right)\right)=\sum _{k=0}^{a-1}{\chi }_{a,N\left(M\right)}\left(k\right)P\left(\frac{x+k}{a}\right).$

The operator ${T}_{{\chi }_{a,N\left(M\right)}}$ satisfies the following properties:

1. (i)

${T}_{{\chi }_{a,N\left(M\right)}}$ is linear on $\mathbb{C}\left[x\right]$ and

${T}_{{\chi }_{a,N\left(M\right)}}:\mathbb{C}\left[x\right]\to \mathbb{C}\left[x\right].$
2. (ii)

${T}_{{\chi }_{a,N\left(M\right)}}$ preserves the degree of the polynomials on $\mathbb{C}\left[x\right]$.

3. (iii)

If we take ${N}_{1}\ge 2$ and ${N}_{2}={N}_{3}=\cdots ={N}_{M}=1$, we have

${T}_{{\chi }_{a,{N}_{1}}}\left(P\left(x\right)\right)=\sum _{k=0}^{a-1}{\xi }^{k}\left({N}_{1}\right)P\left(\frac{x+k}{a}\right).$

Remark 1.2 Setting $N\left(M\right)=\left({N}_{1},1,1,\dots ,1\right)$, ${T}_{{\chi }_{a,\left({N}_{1},1,1,\dots ,1\right)}}$ is reduced to ${T}_{{\chi }_{a,{N}_{1}}}$ (cf. ).

The generating function of generalized Euler-type numbers ${P}_{n,N\left(M\right)}$ is given by

${\mathcal{F}}_{N\left(M\right)}\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}\frac{{t}^{n}}{n!}=\frac{{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)-1}{-1+{e}^{t}{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}$

.

Now, we give the definition of Euler-type polynomials as follows.

Definition 1.3 

The polynomial ${P}_{n,N\left(M\right)}$ is defined by means of the following generating function:

${\mathcal{F}}_{N\left(M\right)}\left(t,x\right)=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}\left(x\right)\frac{{t}^{n}}{n!}=\frac{\left(\left({\prod }_{j=1}^{M}\xi \left({N}_{j}\right)\right)-1\right){e}^{tx}}{\left({\prod }_{j=1}^{M}\xi \left({N}_{j}\right)\right){e}^{t}-1},$
(1)

where

$|t+\sum _{j=1}^{M}\frac{2\pi i}{{N}_{j}}|<2\pi .$

The polynomial ${P}_{n,N\left(M\right)}$ satisfies the following properties:

1. (i)

${P}_{n,N\left(M\right)}\in \mathbb{C}\left[x\right]$.

2. (ii)

${P}_{n,N\left(M\right)}$ is a polynomial with degree n and depends on ${N}_{1},{N}_{2},\dots ,{N}_{M}$.

3. (iii)

If we take ${N}_{1}\ge 2$ and ${N}_{2}={N}_{3}=\cdots ={N}_{M}=1$, we have

$\sum _{n=0}^{\mathrm{\infty }}{P}_{n,{N}_{1}}\left(x\right)\frac{{t}^{n}}{n!}=\frac{\left({\xi }_{{N}_{1}}-1\right){e}^{tx}}{{\xi }_{{N}_{1}}{e}^{t}-1},$

where

$|t+\frac{2\pi i}{{N}_{1}}|<2\pi .$
1. (iv)

We derive the following functional equation:

${\mathcal{F}}_{N\left(M\right)}\left(t,x\right)={\mathcal{F}}_{N\left(M\right)}\left(t\right){e}^{tx},$
(2)

so that, obviously,

${P}_{n,N\left(M\right)}\left(0\right)={P}_{n,N\left(M\right)}.$

We now are ready to define Euler-type numbers and polynomials with order k.

Definition 1.4 Euler-type numbers with order k, ${P}_{n,N\left(M\right)}^{\left(k\right)}$, are defined by means of the following generating functions:

${\mathcal{F}}_{N\left(M\right)}^{\left(k\right)}\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k\right)}\frac{{t}^{n}}{n!},$
(3)

where $k\in \mathbb{N}$ and

$|t+\sum _{j=1}^{M}\frac{2\pi i}{{N}_{j}}|<2\pi .$

Euler-type polynomials with order k are given by the following functional equation:

${\mathcal{F}}_{N\left(M\right)}^{\left(k\right)}\left(t,x\right)={\mathcal{F}}_{N\left(M\right)}^{\left(k\right)}\left(t\right){e}^{tx}=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k\right)}\left(x\right)\frac{{t}^{n}}{n!}.$
(4)

We see that

${\mathcal{F}}_{N\left(M\right)}^{\left(0\right)}\left(t,x\right)={e}^{tx}.$

Thus we obtain

${P}_{n,N\left(M\right)}^{\left(0\right)}\left(x\right)={x}^{n}.$

Remark 1.5 Substituting $k=1$ into (4), we get (2). Therefore, (3) reduces to (1); that is,

${P}_{n,N\left(M\right)}^{\left(1\right)}\left(x\right)={P}_{n,N\left(M\right)}\left(x\right)$

so that, obviously,

${P}_{n,N\left(M\right)}^{\left(1\right)}\left(0\right)={P}_{n,N\left(M\right)}.$

By using (4) and (3), we obtain

$\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k\right)}\left(x\right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k\right)}\frac{{t}^{n}}{n!}\sum _{n=0}^{\mathrm{\infty }}{x}^{n}\frac{{t}^{n}}{n!}.$

Therefore, we get the following theorem.

Theorem 1.6

${P}_{n,N\left(M\right)}^{\left(k\right)}\left(x\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){x}^{n-j}{P}_{j,N\left(M\right)}^{\left(k\right)}.$
(5)

Hence, we arrive at the following definition.

Definition 1.7 Euler-type polynomials with order k, ${P}_{n,N\left(M\right)}^{\left(k\right)}$, are defined by means of the following generating functions:

${\mathcal{F}}_{N\left(M\right)}^{\left(k\right)}\left(t,x\right)=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k\right)}\left(x\right)\frac{{t}^{n}}{n!},$
(6)

where

$|t+\sum _{j=1}^{M}\frac{2\pi i}{{N}_{j}}|<2\pi .$

Note that there is one generating function for each value of k. These are given explicitly as follows:

$\begin{array}{rcl}{\mathcal{F}}_{N\left(M\right)}^{\left(k\right)}\left(t,x\right)& =& {\left(\frac{-1+{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}{-1+{e}^{t}{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}\right)}^{k}{e}^{tx}\\ =& \sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k\right)}\left(x\right)\frac{{t}^{n}}{n!}.\end{array}$

We derive the following functional equation:

${\mathcal{F}}_{N\left(M\right)}^{\left(k+l\right)}\left(t,x\right)={\mathcal{F}}_{N\left(M\right)}^{\left(k\right)}\left(t,x\right){\mathcal{F}}_{N\left(M\right)}^{\left(l\right)}\left(t\right).$
(7)

By using the above functional equation, we arrive at the following theorem.

Theorem 1.8

${P}_{n,N\left(M\right)}^{\left(k+l\right)}\left(x\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){P}_{j,N\left(M\right)}^{\left(k\right)}\left(x\right){P}_{n-j,N\left(M\right)}^{\left(l\right)}.$
(8)

Proof By using (3), (6) and (7), we get

$\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(k+l\right)}\left(x\right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){P}_{j,N\left(M\right)}^{\left(k\right)}\left(x\right){P}_{n-j,N\left(M\right)}^{\left(l\right)}\right)\frac{{t}^{n}}{n!}.$

By comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we get the desired result. □

Substituting $x=0$ into (8), we obtain a convolution formula for the numbers by the following corollary.

Corollary 1.9

${P}_{n,N\left(M\right)}^{\left(k+l\right)}=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){P}_{j,N\left(M\right)}^{\left(k\right)}{P}_{n-j,N\left(M\right)}^{\left(l\right)}.$

By differentiating both sides of equation (2) with respect to the variable x, we obtain the following higher-order differential equation:

$\frac{{\partial }^{j}}{\partial {x}^{j}}{\mathcal{F}}_{N\left(M\right)}\left(t,x\right)={t}^{j}{\mathcal{F}}_{N\left(M\right)}\left(t,x\right).$
(9)

Remark 1.10 Setting $N\left(M\right)=\left({N}_{1},1,1,\dots ,1\right)$, ${P}_{n,\left({N}_{1},1,1,\dots ,1\right)}$ is reduced ${P}_{n,{N}_{1}}\left(x\right)$ (cf. ). Therefore ${P}_{n,N}\left(x\right)$ was defined by generalized Bernoulli-Euler polynomials in  as follows:

$\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N}\left(x\right)\frac{{t}^{n}}{n!}=\left\{\begin{array}{cc}\frac{t{e}^{tx}}{{e}^{t}-1},\hfill & N=1,\hfill \\ \frac{\left({\xi }_{N}-1\right){e}^{tx}}{{\xi }_{N}{e}^{t}-1},\hfill & N\ge 2,\hfill \end{array}$

so that, obviously,

${P}_{n,1}\left(x\right)={B}_{n}\left(x\right)$

and

${P}_{n,2}\left(x\right)={E}_{n}\left(x\right).$

Here ${B}_{n}\left(x\right)$ and ${E}_{n}\left(x\right)$ are Bernoulli polynomials and Euler polynomials, respectively (cf. ).

The Frobenius-Euler polynomial is defined as follows:

Let u be an algebraic number such that $1\ne u\in \mathbb{C}$. Then the Frobenius-Euler polynomial ${H}_{n}\left(x,u\right)$ is defined by

$\frac{1-u}{{e}^{t}-u}{e}^{tx}=\sum _{n=0}^{\mathrm{\infty }}{H}_{n}\left(x,u\right)\frac{{t}^{n}}{n!},$

where

$|t+ln\frac{1}{u}|<2\pi$

(cf. ).

Remark 1.11 Frobenius-Euler number is denoted by ${H}_{n}\left(u\right)$ such that ${H}_{n}\left(0,u\right)={H}_{n}\left(u\right)$. Also, ${H}_{n}\left(x,-1\right)={E}_{n}\left(x\right)$ (cf. ).

By using Frobenius-Euler numbers, one can obtain the Frobenius-Euler polynomials as follows:

${H}_{n}\left(x,u\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){x}^{n-j}{H}_{j}\left(u\right)$

(cf. ).

The Apostol-Bernoulli polynomial is defined as follows.

Definition 1.12 [3, 16]

The Apostol-Bernoulli polynomial ${\mathcal{B}}_{n}\left(x,\lambda \right)$ is defined by

$\frac{t}{\lambda {e}^{t}-1}{e}^{tx}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n}\left(x,\lambda \right)\frac{{t}^{n}}{n!},$

where λ is the arbitrary real or complex parameter and

$|t|<|ln\lambda |.$

Remark 1.13 For $\lambda =1$, we obtain that ${\mathcal{B}}_{n}\left(x,1\right)={B}_{n}\left(x\right)$ (cf. ).

## 2 A functional equation of generalized Euler-type polynomials

Bayad, Aygunes and Simsek showed that for $a\equiv 1mod\left(N\right)$, there exists a unique sequence of monic polynomials ${\left({P}_{n,N}\right)}_{n\in {\mathbb{N}}_{0}}$ in $\mathbb{Q}\left({\xi }_{N}\right)\left[x\right]$ with $deg{P}_{n,N}=n$ such that

${T}_{{\chi }_{a,N}}\left({P}_{n,N}\left(x\right)\right)={a}^{-n}{P}_{n,N}\left(x\right),$

where $a,N\in \mathbb{N}$ (cf. ).

In this section, we give the following theorem.

Theorem 2.1 Let $a,{N}_{1},{N}_{2},\dots ,{N}_{M}\in \mathbb{N}$ and $a\equiv 1\left(mod{N}_{1}{N}_{2}\cdots {N}_{M}\right)$. Then there exists a sequence ${\left({P}_{n,N\left(M\right)}\right)}_{n\in {\mathbb{N}}_{0}}$ in

$\mathbb{Q}\left(\xi \left({N}_{1}\right)\xi \left({N}_{2}\right)\cdots \xi \left({N}_{M}\right)\right)\left[x\right]$

with

$deg{P}_{n,N\left(M\right)}=n$

such that

${T}_{{\chi }_{a,N\left(M\right)}}\left({P}_{n,N\left(M\right)}\left(x\right)\right)={a}^{-n}{P}_{n,N\left(M\right)}\left(x\right).$
(10)

Proof Since ${P}_{n,N\left(M\right)}\in \mathbb{C}\left[x\right]$ and ${T}_{{\chi }_{a,N\left(M\right)}}:\mathbb{C}\left[x\right]\to \mathbb{C}\left[x\right]$, we get

${T}_{{\chi }_{a,N\left(M\right)}}\left({P}_{n,N\left(M\right)}\left(x\right)\right)=\sum _{k=0}^{a-1}{\chi }_{a,N\left(M\right)}\left(k\right){P}_{n,N\left(M\right)}\left(\frac{x+k}{a}\right).$

From the definition of ${\chi }_{a,N\left(M\right)}\left(k\right)$, we have

${T}_{{\chi }_{a,N\left(M\right)}}\left({P}_{n,N\left(M\right)}\left(x\right)\right)=\sum _{k=0}^{a-1}\left(\prod _{j=1}^{M}{e}^{\frac{2\pi ik}{{N}_{j}}}\right){P}_{n,N\left(M\right)}\left(\frac{x+k}{a}\right).$

By using the generating function of ${P}_{n,N\left(M\right)}\left(x\right)$, we get

$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{a-1}\left(\prod _{j=1}^{M}{e}^{\frac{2\pi ik}{{N}_{j}}}\right){P}_{n,N\left(M\right)}\left(\frac{x+k}{a}\right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{k=0}^{a-1}\left(\prod _{j=1}^{M}{e}^{\frac{2\pi ik}{{N}_{j}}}\right)\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}\left(\frac{x+k}{a}\right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{k=0}^{a-1}\left(\prod _{j=1}^{M}{e}^{\frac{2\pi ik}{{N}_{j}}}\right)\frac{\left(\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right)-1\right){e}^{t\left(\frac{x+k}{a}\right)}}{\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right){e}^{t}-1}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{\left(\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right)-1\right){e}^{\frac{tx}{a}}}{\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right){e}^{t}-1}\sum _{k=0}^{a-1}\left(exp\left(\sum _{j=1}^{M}{e}^{\frac{2\pi ik}{{N}_{j}}}\right)\right)exp\left(\frac{tk}{a}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{\left(\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right)-1\right){e}^{\frac{tx}{a}}}{\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right){e}^{t}-1}\sum _{k=0}^{a-1}{\left(exp\left(\frac{t}{a}+\sum _{j=1}^{M}\frac{2\pi i}{{N}_{j}}\right)\right)}^{k}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{\left(\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right)-1\right){e}^{\frac{tx}{a}}}{\left({\prod }_{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}\right){e}^{t}-1}\frac{{e}^{t}{\left(exp\left({\sum }_{j=1}^{M}\frac{2\pi i}{{N}_{j}}\right)\right)}^{a}-1}{{e}^{\frac{t}{a}}\left(exp\left({\sum }_{j=1}^{M}\frac{2\pi i}{{N}_{j}}\right)\right)-1}.\hfill \end{array}$

Since $a\equiv 1\left(mod{N}_{1}{N}_{2}\cdots {N}_{M}\right)$, the following relation holds:

${\left(exp\left(\sum _{j=1}^{M}\frac{2\pi i}{{N}_{j}}\right)\right)}^{a}=exp\left(\sum _{j=1}^{M}\frac{2\pi i}{{N}_{j}}\right)=\prod _{j=1}^{M}{e}^{\frac{2\pi i}{{N}_{j}}}.$

Therefore, we have

$\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{a-1}\left(\prod _{j=1}^{M}{e}^{\frac{2\pi ik}{{N}_{j}}}\right){P}_{n,N\left(M\right)}\left(\frac{x+k}{a}\right)\right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}{a}^{-n}{P}_{n,N\left(M\right)}\left(x\right)\frac{{t}^{n}}{n!}.$

By comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we get the desired result. □

Remark 2.2 A different proof of (10) is given in . If we take ${N}_{1}\ge 2$ and ${N}_{2}={N}_{3}=\cdots ={N}_{M}=1$, we have the following functional equation:

${T}_{{\chi }_{a,{N}_{1}}}\left({P}_{n,{N}_{1}}\left(x\right)\right)={a}^{-n}{P}_{n,{N}_{1}}\left(x\right)$

which is satisfied for generalized Bernoulli-Euler polynomials in .

## 3 Some properties of generalized Euler-type polynomials

In this section, we obtain some relations between generalized Euler-type polynomials, Apostol-Bernoulli polynomials and Frobenius-Euler polynomials. Also, we give a formula to obtain the generalized Euler-type polynomials.

Theorem 3.1 Let $n\in \mathbb{N}$. Then we have

${P}_{n+1,N\left(M\right)}\left(x\right)={P}_{n,N\left(M\right)}\left(x\right)+\frac{{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}{1-{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){P}_{k,N\left(M\right)}^{\left(2\right)}\left(x\right).$

Proof By differentiating both sides of equation (2) with respect to the variable t, we have

$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{P}_{n+1,N\left(M\right)}\left(x\right)\frac{{t}^{n}}{n!}& =& \frac{\partial }{\partial t}{\mathcal{F}}_{N\left(M\right)}\left(t,x\right)\\ =& {\mathcal{F}}_{N\left(M\right)}\left(t,x\right)+\left(\frac{{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}{1-{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}\right){e}^{t}{e}^{tx}{\left({\mathcal{F}}_{N\left(M\right)}\left(t\right)\right)}^{2}\\ =& \sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}\left(x\right)\frac{{t}^{n}}{n!}+\left(\frac{{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}{1-{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}\right){e}^{t}\left(\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}^{\left(2\right)}\left(x\right)\frac{{t}^{n}}{n!}\right).\end{array}$

Therefore, we obtain

$\sum _{n=0}^{\mathrm{\infty }}{P}_{n+1,N\left(M\right)}\left(x\right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}\left({P}_{n,N\left(M\right)}\left(x\right)+\frac{{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}{1-{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){P}_{k,N\left(M\right)}^{\left(2\right)}\left(x\right)\right)\frac{{t}^{n}}{n!}.$

By comparing the coefficients of $\frac{{t}^{n}}{n!}$, we obtain the desired result. □

In the following theorem, we give a relation between the polynomials ${P}_{n,N\left(M\right)}\left(x\right)$ and Frobenius-Euler polynomials.

Theorem 3.2 

Let $n\in {\mathbb{N}}_{0}$. Then we have

${P}_{n,N\left(M\right)}\left(x\right)={H}_{n}\left(x,\prod _{j=1}^{M}\frac{1}{\xi \left({N}_{j}\right)}\right).$

Proof By using the generating function of ${P}_{n,N\left(M\right)}\left(x\right)$, we have

$\sum _{n=0}^{\mathrm{\infty }}{P}_{n,N\left(M\right)}\left(x\right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}{H}_{n}\left(x,\prod _{j=1}^{M}\frac{1}{\xi \left({N}_{j}\right)}\right)\frac{{t}^{n}}{n!}.$

In the above equation, if we compare the coefficients of $\frac{{t}^{n}}{n!}$, we get the desired result. □

In the following theorem, we give a relation between ${P}_{n,N\left(M\right)}\left(x\right)$ and Apostol-Bernoulli polynomials.

Theorem 3.3 

Let $n\in \mathbb{N}$. Then we have

${P}_{n-1,N\left(M\right)}\left(x\right)=\left(\prod _{j=1}^{M}\xi \left({N}_{j}\right)-1\right)\frac{1}{n}{\mathcal{B}}_{n}\left(x,\prod _{j=1}^{M}\xi \left({N}_{j}\right)\right).$

Proof

We arrange the generating function of generalized Euler-type polynomials as follows:

$\sum _{n=1}^{\mathrm{\infty }}{P}_{n-1,N\left(M\right)}\frac{{t}^{n-1}}{\left(n-1\right)!}=\frac{{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)-1}{{e}^{t}{\prod }_{j=1}^{M}\xi \left({N}_{j}\right)-1}{e}^{xt}.$

Therefore, we have

$\sum _{n=1}^{\mathrm{\infty }}{P}_{n-1,N\left(M\right)}\frac{{t}^{n-1}}{\left(n-1\right)!}=\sum _{n=1}^{\mathrm{\infty }}\left(\frac{1}{n}\left(\prod _{j=1}^{M}\xi \left({N}_{j}\right)-1\right){\mathcal{B}}_{n}\left(x,\prod _{j=1}^{M}\xi \left({N}_{j}\right)\right)\right)\frac{{t}^{n-1}}{\left(n-1\right)!}.$

In the above equation, if we compare the coefficients of $\frac{{t}^{n-1}}{\left(n-1\right)!}$, we get the desired result. □

In the following theorem, it is possible to find the generalized Euler-type polynomials.

Theorem 3.4 Let $n\in {\mathbb{N}}_{0}$. Then we have

${P}_{n,N\left(M\right)}\left(x\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){x}^{n-j}{P}_{j,N\left(M\right)}.$
(11)

Proof of (11) is the same as that of (5), so we omit it .

$\begin{array}{c}{P}_{1,N\left(M\right)}=\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1},\hfill \\ {P}_{2,N\left(M\right)}=\frac{2}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1},\hfill \\ {P}_{3,N\left(M\right)}=\frac{6}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{3}}+\frac{6}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1}\hfill \end{array}$

and

${P}_{4,N\left(M\right)}=\frac{24}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{4}}+\frac{36}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{3}}+\frac{14}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1}.$

By using (11), we have the following list for the generalized Euler-type polynomials:

$\begin{array}{c}{P}_{0,N\left(M\right)}\left(x\right)=1,\hfill \\ {P}_{1,N\left(M\right)}\left(x\right)=x+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1},\hfill \\ {P}_{2,N\left(M\right)}\left(x\right)={x}^{2}+x\left(\frac{2}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)+\left(\frac{2}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right),\hfill \\ \begin{array}{rl}{P}_{3,N\left(M\right)}\left(x\right)=& {x}^{3}+{x}^{2}\left(\frac{3}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)+x\left(\frac{6}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{3}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)\\ +\left(\frac{6}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{3}}+\frac{6}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)\end{array}\hfill \end{array}$

and

$\begin{array}{rcl}{P}_{4,N\left(M\right)}\left(x\right)& =& {x}^{4}+{x}^{3}\left(\frac{4}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)+{x}^{2}\left(\frac{12}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{6}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)\\ +x\left(\frac{24}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{3}}+\frac{24}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{4}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right)\\ +\left(\frac{24}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{4}}+\frac{36}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{3}}+\frac{14}{{\left({\chi }_{a,N\left(M\right)}^{-1}-1\right)}^{2}}+\frac{1}{{\chi }_{a,N\left(M\right)}^{-1}-1}\right).\end{array}$

## Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

## Author information

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Correspondence to Aykut Ahmet Aygunes.

### Competing interests

The author declares that he has no competing interests.

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Aygunes, A.A. Higher-order Euler-type polynomials and their applications. Fixed Point Theory Appl 2013, 40 (2013). https://doi.org/10.1186/1687-1812-2013-40

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• DOI: https://doi.org/10.1186/1687-1812-2013-40

### Keywords

• generalized partial Hecke operators
• higher-order Euler-type polynomials
• higher-order Euler-type numbers
• Apostol-Bernoulli polynomials
• Frobenius-Euler polynomials
• Euler polynomials
• Euler numbers
• functional equation
• generating functions 