Theory and Modern Applications

# Apostol-Euler polynomials arising from umbral calculus

## Abstract

In this paper, by using the orthogonality type as defined in the umbral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the Apostol-Euler polynomials.

MSC:05A40.

## 1 Introduction

Let be the set of all polynomials in a single variable x over the complex field â„‚ of degree at most n. Clearly, is a $\left(n+1\right)$-dimensional vector space over â„‚. Define

$\mathcal{H}=\left\{f\left(t\right)=\underset{kâ‰¥0}{âˆ‘}{a}_{k}\frac{{t}^{k}}{k!}|{a}_{k}âˆˆ\mathbb{C}\right\}$
(1.1)

to be the algebra of formal power series in a single variable t. As is known, $ã€ˆL|p\left(x\right)ã€‰$ denotes the action of a linear functional $Lâˆˆ\mathcal{H}$ on a polynomial $p\left(x\right)$, and we remind that the vector space on is defined by

$ã€ˆcL+{c}^{â€²}{L}^{â€²}|p\left(x\right)ã€‰=cã€ˆL|p\left(x\right)ã€‰+{c}^{â€²}ã€ˆ{L}^{â€²}|p\left(x\right)ã€‰$

for any $c,{c}^{â€²}âˆˆ\mathbb{C}$ and $L,{L}^{â€²}âˆˆ\mathcal{H}$ (see [1â€“4]). The formal power series in variable t define a linear functional on by setting

(1.2)

By (1.1) and (1.2), we have

(1.3)

where ${\mathrm{Î´}}_{n,k}$ is the Kronecker symbol. Let ${f}_{L}\left(t\right)={âˆ‘}_{kâ‰¥0}ã€ˆL|{x}^{k}ã€‰\frac{{t}^{k}}{k!}$ with $Lâˆˆ\mathcal{H}$. From (1.3), we have $ã€ˆ{f}_{L}\left(t\right)|{x}^{n}ã€‰=ã€ˆL|{x}^{n}ã€‰$. So, the map $Lâ†¦{f}_{L}\left(t\right)$ is a vector space isomorphic from onto â„‹. Henceforth, â„‹ is thought of as a set of both formal power series and linear functionals. We call â„‹ umbral algebra. The umbral calculus is the study of umbral algebra.

Let $f\left(t\right)âˆˆ\mathcal{H}$. The smallest integer k for which the coefficient of ${t}^{k}$ does not vanish is called the order of $f\left(t\right)$ and is denoted by $O\left(f\left(t\right)\right)$ (see [1â€“4]). If $O\left(f\left(t\right)\right)=1$, $O\left(f\left(t\right)\right)=0$, then $f\left(t\right)$ is called a delta, an invertible series, respectively. For given two power series $f\left(t\right),g\left(t\right)âˆˆ\mathcal{H}$ such that $O\left(f\left(t\right)\right)=1$ and $O\left(g\left(t\right)\right)=0$, there exists a unique sequence ${S}_{n}\left(x\right)$ of polynomials with $ã€ˆg\left(t\right){\left(f\left(t\right)\right)}^{k}|{S}_{n}\left(x\right)ã€‰=n!{\mathrm{Î´}}_{n,k}$ (this condition sometimes is called orthogonality type) for all $n,kâ‰¥0$. The sequence ${S}_{n}\left(x\right)$ is called the Sheffer sequence for $\left(g\left(t\right),f\left(t\right)\right)$ which is denoted by ${S}_{n}\left(x\right)âˆ¼\left(g\left(t\right),f\left(t\right)\right)$ (see [1â€“4]).

For $f\left(t\right)âˆˆ\mathcal{H}$ and , we have

$ã€ˆ{e}^{yt}|p\left(x\right)ã€‰=p\left(y\right),\phantom{\rule{2em}{0ex}}ã€ˆf\left(t\right)g\left(t\right)|p\left(x\right)ã€‰=ã€ˆf\left(t\right)|g\left(t\right)p\left(x\right)ã€‰,$
(1.4)

and

$f\left(t\right)=\underset{kâ‰¥0}{âˆ‘}ã€ˆf\left(t\right)|{x}^{k}ã€‰\frac{{t}^{k}}{k!},\phantom{\rule{2em}{0ex}}p\left(x\right)=\underset{kâ‰¥0}{âˆ‘}ã€ˆ{t}^{k}|p\left(x\right)ã€‰\frac{{x}^{k}}{k!}$
(1.5)

(see [1â€“4]). From (1.5), we derive

$ã€ˆ{t}^{k}|p\left(x\right)ã€‰={p}^{\left(k\right)}\left(0\right),\phantom{\rule{2em}{0ex}}ã€ˆ1|{p}^{\left(k\right)}\left(x\right)ã€‰={p}^{\left(k\right)}\left(0\right),$
(1.6)

where ${p}^{\left(k\right)}\left(0\right)$ denotes the k th derivative of $p\left(x\right)$ with respect to x at $x=0$. Let ${S}_{n}\left(x\right)âˆ¼\left(g\left(t\right),f\left(t\right)\right)$. Then we have

$\frac{1}{g\left(\stackrel{Â¯}{f}\left(t\right)\right)}{e}^{y\stackrel{Â¯}{f}\left(t\right)}=\underset{kâ‰¥0}{âˆ‘}{S}_{k}\left(y\right)\frac{{t}^{k}}{k!},$
(1.7)

for all $yâˆˆ\mathbb{C}$, where $\stackrel{Â¯}{f}\left(t\right)$ is the compositional inverse of $f\left(t\right)$ (see [1â€“6]).

For $\mathrm{Î»}âˆˆ\mathbb{C}$ with , the Apostol-Euler polynomials (see [7â€“10]) are defined by the generating function to be

$\frac{2}{\mathrm{Î»}{e}^{t}+1}{e}^{xt}=\underset{kâ‰¥0}{âˆ‘}{E}_{k}\left(x|\mathrm{Î»}\right)\frac{{t}^{k}}{k!}.$
(1.8)

In particular, $x=0$, ${E}_{n}\left(0|\mathrm{Î»}\right)={E}_{n}\left(\mathrm{Î»}\right)$ is called the nth Apostol-Euler number. From (1.8), we can derive

${E}_{n}\left(x|\mathrm{Î»}\right)=\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{nâˆ’k}\left(\mathrm{Î»}\right){x}^{k}.$
(1.9)

By (1.9), we have $\frac{d}{dx}{E}_{n}\left(x|\mathrm{Î»}\right)=n{E}_{nâˆ’1}\left(x|\mathrm{Î»}\right)$. Also, from (1.8) we have

$\frac{2}{\mathrm{Î»}{e}^{t}+1}={e}^{E\left(\mathrm{Î»}\right)t}=\underset{nâ‰¥0}{âˆ‘}{E}_{n}\left(\mathrm{Î»}\right)\frac{{t}^{n}}{n!}$
(1.10)

with the usual convention about replacing ${E}^{n}\left(\mathrm{Î»}\right)$ by ${E}_{n}\left(\mathrm{Î»}\right)$. By (1.10), we get

$2={e}^{E\left(\mathrm{Î»}\right)t}\left(\mathrm{Î»}{e}^{t}+1\right)=\mathrm{Î»}{e}^{\left(E\left(\mathrm{Î»}\right)+1\right)t}+{e}^{E\left(\mathrm{Î»}\right)t}=\underset{nâ‰¥0}{âˆ‘}\left(\mathrm{Î»}{\left(E\left(\mathrm{Î»}\right)+1\right)}^{n}+{E}_{n}\left(\mathrm{Î»}\right)\right)\frac{{t}^{n}}{n!}.$

Thus, by comparing the coefficients of the both sides, we have

$\mathrm{Î»}{\left(E\left(\mathrm{Î»}\right)+1\right)}^{n}+{E}_{n}\left(\mathrm{Î»}\right)=2{\mathrm{Î´}}_{n,0}.$
(1.11)

As is well known, the Bernoulli polynomial (see [11â€“14]) is also defined by the generating function to be

$\frac{t}{{e}^{t}âˆ’1}{e}^{xt}=\underset{kâ‰¥0}{âˆ‘}{B}_{k}\left(x\right)\frac{{t}^{k}}{k!}.$
(1.12)

In the special case, $x=0$, ${B}_{n}\left(0\right)={B}_{n}$ is called the nth Bernoulli number. By (1.12), we get

${B}_{n}\left(x\right)=\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){B}_{nâˆ’k}{x}^{k}.$
(1.13)

From (1.12), we note that

$\frac{t}{{e}^{t}âˆ’1}={e}^{Bt}=\underset{nâ‰¥0}{âˆ‘}{B}_{n}\frac{{t}^{n}}{n!}$
(1.14)

with the usual convention about replacing ${B}^{n}$ by ${B}_{n}$. By (1.13) and (1.14), we get

$t={e}^{Bt}\left({e}^{t}âˆ’1\right)={e}^{\left(B+1\right)t}âˆ’{e}^{Bt}=\underset{nâ‰¥0}{âˆ‘}\left({\left(B+1\right)}^{n}âˆ’{B}_{n}\right)\frac{{t}^{n}}{n!},$

which implies

${B}_{n}\left(1\right)âˆ’{B}_{n}={\left(B+1\right)}^{n}âˆ’{B}_{n}={\mathrm{Î´}}_{n,1},\phantom{\rule{2em}{0ex}}{B}_{0}=1.$
(1.15)

Euler polynomials (see [4, 11, 13, 15]) are defined by

$\frac{2}{{e}^{t}+1}{e}^{xt}=\underset{kâ‰¥0}{âˆ‘}{E}_{k}\left(x\right)\frac{{t}^{k}}{k!}.$
(1.16)

In the special case, $x=0$, ${E}_{n}\left(0\right)={E}_{n}$ is called the nth Euler number. By (1.16), we get

$\frac{2}{{e}^{t}+1}={e}^{Et}=\underset{nâ‰¥0}{âˆ‘}{E}_{n}\frac{{t}^{n}}{n!}$
(1.17)

with the usual convention about replacing ${E}^{n}$ by ${E}_{n}$. By (1.16) and (1.17), we get

$2={e}^{Et}\left({e}^{t}+1\right)={e}^{\left(E+1\right)t}+{e}^{Et}=\underset{nâ‰¥0}{âˆ‘}\left({\left(E+1\right)}^{n}+{E}_{n}\right)\frac{{t}^{n}}{n!},$

which implies

${E}_{n}\left(1\right)+{E}_{n}={\left(E+1\right)}^{n}+{E}_{n}=2{\mathrm{Î´}}_{n,0}.$
(1.18)

For $\mathrm{Î»}âˆˆ\mathbb{C}$ with , the Frobenius-Euler (see [11, 16â€“19]) polynomials are defined by

$\frac{1+\mathrm{Î»}}{{e}^{t}+\mathrm{Î»}}{e}^{xt}=\underset{kâ‰¥0}{âˆ‘}{F}_{k}\left(x|âˆ’\mathrm{Î»}\right)\frac{{t}^{k}}{k!}.$
(1.19)

In the special case, $x=0$, ${F}_{n}\left(0|âˆ’\mathrm{Î»}\right)={F}_{n}\left(âˆ’\mathrm{Î»}\right)$ is called the nth Frobenius-Euler number (see [17]). By (1.19), we get

$\frac{1+\mathrm{Î»}}{{e}^{t}+\mathrm{Î»}}={e}^{Ft}=\underset{nâ‰¥0}{âˆ‘}{F}_{n}\left(âˆ’\mathrm{Î»}\right)\frac{{t}^{n}}{n!}$
(1.20)

with the usual convention about replacing ${F}^{n}\left(âˆ’\mathrm{Î»}\right)$ by ${F}_{n}\left(âˆ’\mathrm{Î»}\right)$ (see [17]). By (1.19) and (1.20), we get

$1+\mathrm{Î»}={e}^{F\left(âˆ’\mathrm{Î»}\right)t}\left({e}^{t}+\mathrm{Î»}\right)={e}^{\left(F\left(âˆ’\mathrm{Î»}\right)+1\right)t}+\mathrm{Î»}{e}^{F\left(âˆ’\mathrm{Î»}\right)t}=\underset{nâ‰¥0}{âˆ‘}\left({\left(F\left(âˆ’\mathrm{Î»}\right)+1\right)}^{n}+\mathrm{Î»}{F}_{n}\left(âˆ’\mathrm{Î»}\right)\right)\frac{{t}^{n}}{n!},$

which implies

$\mathrm{Î»}{F}_{n}\left(âˆ’\mathrm{Î»}\right)+{F}_{n}\left(1|âˆ’\mathrm{Î»}\right)=\mathrm{Î»}{F}_{n}\left(âˆ’\mathrm{Î»}\right)+{\left(F\left(âˆ’\mathrm{Î»}\right)+1\right)}^{n}=\left(1+\mathrm{Î»}\right){\mathrm{Î´}}_{n,0}.$
(1.21)

In the next section, we present our main theorem and its applications. More precisely, by using the orthogonality type, we write any polynomial in as a linear combination of the Apostol-Euler polynomials. Several applications related to Bernoulli, Euler and Frobenius-Euler polynomials are derived.

## 2 Main results and applications

Note that the set of the polynomials ${E}_{0}\left(x|\mathrm{Î»}\right),{E}_{1}\left(x|\mathrm{Î»}\right),â€¦,{E}_{n}\left(x|\mathrm{Î»}\right)$ is a good basis for . Thus, for , there exist constants ${c}_{0},{c}_{1},â€¦,{c}_{n}$ such that $p\left(x\right)={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$. Since ${E}_{n}\left(x|\mathrm{Î»}\right)âˆ¼\left(\left(1+\mathrm{Î»}{e}^{t}\right)/2,t\right)$ (see (1.7) and (1.8)), we have

$ã€ˆ\frac{1+\mathrm{Î»}{e}^{t}}{2}{t}^{k}|{E}_{n}\left(x|\mathrm{Î»}\right)ã€‰=n!{\mathrm{Î´}}_{n,k},$

which gives

$ã€ˆ\frac{1+\mathrm{Î»}{e}^{t}}{2}{t}^{k}|p\left(x\right)ã€‰=\underset{\mathrm{â„“}=0}{\overset{n}{âˆ‘}}{c}_{\mathrm{â„“}}ã€ˆ\frac{1+\mathrm{Î»}{e}^{t}}{2}{t}^{k}|{E}_{\mathrm{â„“}}\left(x|\mathrm{Î»}\right)ã€‰=\underset{\mathrm{â„“}=0}{\overset{n}{âˆ‘}}{c}_{\mathrm{â„“}}\mathrm{â„“}!{\mathrm{Î´}}_{\mathrm{â„“},k}=k!{c}_{k}.$

Hence, we can state the following result.

Theorem 2.1 For all , there exist constants ${c}_{0},{c}_{1},â€¦,{c}_{n}$ such that $p\left(x\right)={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$, where

${c}_{k}=\frac{1}{2k!}ã€ˆ\left(1+\mathrm{Î»}{e}^{t}\right){t}^{k}|p\left(x\right)ã€‰.$

Now, we present several applications for our theorem. As a first application, let us take $p\left(x\right)={x}^{n}$ with $nâ‰¥0$. By Theorem 2.1, we have ${x}^{n}={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$, where

${c}_{k}=\frac{1}{2k!}ã€ˆ\left(1+\mathrm{Î»}{e}^{t}\right){t}^{k}|{x}^{n}ã€‰=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)ã€ˆ1+\mathrm{Î»}{e}^{t}|{x}^{nâˆ’k}ã€‰=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({\mathrm{Î´}}_{nâˆ’k,0}+\mathrm{Î»}\right),$

which implies the following identity.

Corollary 2.2 For all $nâ‰¥0$,

${x}^{n}=\frac{1}{2}{E}_{n}\left(x|\mathrm{Î»}\right)+\frac{\mathrm{Î»}}{2}\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{k}\left(x|\mathrm{Î»}\right).$

Let , then by Theorem 2.1 we have that ${B}_{n}\left(x\right)={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$, where

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}ã€ˆ\left(1+\mathrm{Î»}{e}^{t}\right){t}^{k}|{B}_{n}\left(x\right)ã€‰=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)ã€ˆ1+\mathrm{Î»}{e}^{t}|{B}_{nâˆ’k}\left(x\right)ã€‰\\ =\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({B}_{nâˆ’k}+\mathrm{Î»}{B}_{nâˆ’k}\left(1\right)\right),\end{array}$

which, by (1.15), implies the following identity.

Corollary 2.3 For all $nâ‰¥2$,

Let $p\left(x\right)={E}_{n}\left(x\right)$, then by Theorem 2.1 we have that ${E}_{n}\left(x\right)={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$, where

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}ã€ˆ\left(1+\mathrm{Î»}{e}^{t}\right){t}^{k}|{E}_{n}\left(x\right)ã€‰=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)ã€ˆ1+\mathrm{Î»}{e}^{t}|{E}_{nâˆ’k}\left(x\right)ã€‰\\ =\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({E}_{nâˆ’k}+\mathrm{Î»}{E}_{nâˆ’k}\left(1\right)\right),\end{array}$

which, by (1.18), implies the following identity.

Corollary 2.4 For all $nâ‰¥0$,

${E}_{n}\left(x\right)=\frac{1+\mathrm{Î»}}{2}\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{nâˆ’k}{E}_{k}\left(x|\mathrm{Î»}\right).$

For another application, let $p\left(x\right)={F}_{n}\left(x|âˆ’\mathrm{Î»}\right)$, then by Theorem 2.1 we have that ${F}_{n}\left(x|âˆ’\mathrm{Î»}\right)={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$, where

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}ã€ˆ\left(1+\mathrm{Î»}{e}^{t}\right){t}^{k}|{F}_{n}\left(x|âˆ’\mathrm{Î»}\right)ã€‰=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)ã€ˆ1+\mathrm{Î»}{e}^{t}|{F}_{nâˆ’k}\left(x|âˆ’\mathrm{Î»}\right)ã€‰\\ =\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({F}_{nâˆ’k}\left(âˆ’\mathrm{Î»}\right)+\mathrm{Î»}{F}_{nâˆ’k}\left(1|âˆ’\mathrm{Î»}\right)\right),\end{array}$

which, by (1.21), implies the following identity.

Corollary 2.5 For all $nâ‰¥1$,

${F}_{n}\left(x|âˆ’\mathrm{Î»}\right)=\frac{1+\mathrm{Î»}}{2}{E}_{n}\left(x|\mathrm{Î»}\right)+\frac{1âˆ’{\mathrm{Î»}}^{2}}{2}\underset{k=0}{\overset{nâˆ’1}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){F}_{nâˆ’k}\left(âˆ’\mathrm{Î»}\right){E}_{k}\left(x|\mathrm{Î»}\right).$

Again, let $p\left(x\right)={y}_{n}\left(x\right)={âˆ‘}_{k=0}^{n}\frac{\left(n+k\right)!}{\left(nâˆ’k\right)!k!}\frac{{x}^{k}}{{2}^{k}}$ be the n th Bessel polynomial (which is the solution of the following differential equation ${x}^{2}{f}^{â€³}\left(x\right)+2\left(x+1\right){f}^{â€²}+n\left(n+1\right)f=0$, where ${f}^{â€²}\left(x\right)$ denotes the derivative of $f\left(x\right)$, see [3, 4]). Then, by Theorem 2.1, we can write ${y}_{n}\left(x\right)={âˆ‘}_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\mathrm{Î»}\right)$, where

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}\underset{\mathrm{â„“}=0}{\overset{n}{âˆ‘}}\frac{\left(n+\mathrm{â„“}\right)!}{\left(nâˆ’\mathrm{â„“}\right)!\mathrm{â„“}!{2}^{\mathrm{â„“}}}ã€ˆ1+\mathrm{Î»}{e}^{t}|{t}^{k}{x}^{\mathrm{â„“}}ã€‰\\ =\frac{1}{2}\underset{\mathrm{â„“}=k}{\overset{n}{âˆ‘}}\frac{\left(n+\mathrm{â„“}\right)!}{\left(nâˆ’\mathrm{â„“}\right)!\mathrm{â„“}!{2}^{\mathrm{â„“}}}\left(\genfrac{}{}{0}{}{\mathrm{â„“}}{k}\right)ã€ˆ1+\mathrm{Î»}{e}^{t}|{x}^{\mathrm{â„“}âˆ’k}ã€‰\\ =\frac{1}{2}\underset{\mathrm{â„“}=k}{\overset{n}{âˆ‘}}\frac{\left(n+\mathrm{â„“}\right)!}{\left(nâˆ’\mathrm{â„“}\right)!\mathrm{â„“}!{2}^{\mathrm{â„“}}}\left(\genfrac{}{}{0}{}{\mathrm{â„“}}{k}\right)\left({\mathrm{Î´}}_{nâˆ’k,0}+\mathrm{Î»}\right)\\ =\frac{k!}{{2}^{k+1}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+k}{k}\right)+\mathrm{Î»}\underset{\mathrm{â„“}=k}{\overset{n}{âˆ‘}}\frac{k!}{{2}^{\mathrm{â„“}+1}}\left(\genfrac{}{}{0}{}{\mathrm{â„“}}{k}\right)\left(\genfrac{}{}{0}{}{n}{\mathrm{â„“}}\right)\left(\genfrac{}{}{0}{}{n+\mathrm{â„“}}{\mathrm{â„“}}\right),\end{array}$

which implies the following identity.

Corollary 2.6 For all $nâ‰¥1$,

${y}_{n}\left(x\right)=\underset{k=0}{\overset{n}{âˆ‘}}\frac{k!}{{2}^{k+1}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+k}{k}\right){E}_{k}\left(x|\mathrm{Î»}\right)+\mathrm{Î»}\underset{k=0}{\overset{n}{âˆ‘}}\underset{\mathrm{â„“}=k}{\overset{n}{âˆ‘}}\frac{k!}{{2}^{\mathrm{â„“}+1}}\left(\genfrac{}{}{0}{}{\mathrm{â„“}}{k}\right)\left(\genfrac{}{}{0}{}{n}{\mathrm{â„“}}\right)\left(\genfrac{}{}{0}{}{n+\mathrm{â„“}}{\mathrm{â„“}}\right){E}_{k}\left(x|\mathrm{Î»}\right).$

We end by noting that if we substitute $\mathrm{Î»}=0$ in any of our corollaries, then we get the well-known value of the polynomial $p\left(x\right)$. For instance, by setting $\mathrm{Î»}=0$, the last corollary gives that ${y}_{n}\left(x\right)={âˆ‘}_{k=0}^{n}\frac{\left(n+k\right)!}{\left(nâˆ’k\right)!k!}\frac{{x}^{k}}{{2}^{k}}$, as expected.

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## Acknowledgements

This paper is supported in part by the Research Grant of Kwangwoon University in 2013.

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Correspondence to Taekyun Kim.

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The authors declare that they have no competing interests.

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All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Kim, T., Mansour, T., Rim, SH. et al. Apostol-Euler polynomials arising from umbral calculus. Adv Differ Equ 2013, 301 (2013). https://doi.org/10.1186/1687-1847-2013-301