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 ${\mathrm{\Pi }}_{n}$ be the set of all polynomials in a single variable x over the complex field of degree at most n. Clearly, ${\mathrm{\Pi }}_{n}$ is a $\left(n+1\right)$-dimensional vector space over . Define

$\mathcal{H}=\left\{f\left(t\right)=\sum _{k\ge 0}{a}_{k}\frac{{t}^{k}}{k!}|{a}_{k}\in \mathbb{C}\right\}$
(1.1)

to be the algebra of formal power series in a single variable t. As is known, $〈L|p\left(x\right)〉$ denotes the action of a linear functional $L\in \mathcal{H}$ on a polynomial $p\left(x\right)$, and we remind that the vector space on ${\mathrm{\Pi }}_{n}$ is defined by

$〈cL+{c}^{\prime }{L}^{\prime }|p\left(x\right)〉=c〈L|p\left(x\right)〉+{c}^{\prime }〈{L}^{\prime }|p\left(x\right)〉$

for any $c,{c}^{\prime }\in \mathbb{C}$ and $L,{L}^{\prime }\in \mathcal{H}$ (see ). The formal power series in variable t define a linear functional on ${\mathrm{\Pi }}_{n}$ by setting

(1.2)

By (1.1) and (1.2), we have

(1.3)

where ${\delta }_{n,k}$ is the Kronecker symbol. Let ${f}_{L}\left(t\right)={\sum }_{k\ge 0}〈L|{x}^{k}〉\frac{{t}^{k}}{k!}$ with $L\in \mathcal{H}$. From (1.3), we have $〈{f}_{L}\left(t\right)|{x}^{n}〉=〈L|{x}^{n}〉$. So, the map $L↦{f}_{L}\left(t\right)$ is a vector space isomorphic from ${\mathrm{\Pi }}_{n}$ onto . Henceforth, is thought of as a set of both formal power series and linear functionals. We call umbral algebra. The umbral calculus is the study of umbral algebra.

Let $f\left(t\right)\in \mathcal{H}$. 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 ). 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 $f\left(t\right),g\left(t\right)\in \mathcal{H}$ 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 $〈g\left(t\right){\left(f\left(t\right)\right)}^{k}|{S}_{n}\left(x\right)〉=n!{\delta }_{n,k}$ (this condition sometimes is called orthogonality type) for all $n,k\ge 0$. 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 ${S}_{n}\left(x\right)\sim \left(g\left(t\right),f\left(t\right)\right)$ (see ).

For $f\left(t\right)\in \mathcal{H}$ and $p\left(x\right)\in \mathrm{\Pi }$, we have

$〈{e}^{yt}|p\left(x\right)〉=p\left(y\right),\phantom{\rule{2em}{0ex}}〈f\left(t\right)g\left(t\right)|p\left(x\right)〉=〈f\left(t\right)|g\left(t\right)p\left(x\right)〉,$
(1.4)

and

$f\left(t\right)=\sum _{k\ge 0}〈f\left(t\right)|{x}^{k}〉\frac{{t}^{k}}{k!},\phantom{\rule{2em}{0ex}}p\left(x\right)=\sum _{k\ge 0}〈{t}^{k}|p\left(x\right)〉\frac{{x}^{k}}{k!}$
(1.5)

(see ). From (1.5), we derive

$〈{t}^{k}|p\left(x\right)〉={p}^{\left(k\right)}\left(0\right),\phantom{\rule{2em}{0ex}}〈1|{p}^{\left(k\right)}\left(x\right)〉={p}^{\left(k\right)}\left(0\right),$
(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 ${S}_{n}\left(x\right)\sim \left(g\left(t\right),f\left(t\right)\right)$. Then we have

$\frac{1}{g\left(\overline{f}\left(t\right)\right)}{e}^{y\overline{f}\left(t\right)}=\sum _{k\ge 0}{S}_{k}\left(y\right)\frac{{t}^{k}}{k!},$
(1.7)

for all $y\in \mathbb{C}$, where $\overline{f}\left(t\right)$ is the compositional inverse of $f\left(t\right)$ (see ).

For $\lambda \in \mathbb{C}$ with $\lambda \ne -1$, the Apostol-Euler polynomials (see ) are defined by the generating function to be

$\frac{2}{\lambda {e}^{t}+1}{e}^{xt}=\sum _{k\ge 0}{E}_{k}\left(x|\lambda \right)\frac{{t}^{k}}{k!}.$
(1.8)

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

${E}_{n}\left(x|\lambda \right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{n-k}\left(\lambda \right){x}^{k}.$
(1.9)

By (1.9), we have $\frac{d}{dx}{E}_{n}\left(x|\lambda \right)=n{E}_{n-1}\left(x|\lambda \right)$. Also, from (1.8) we have

$\frac{2}{\lambda {e}^{t}+1}={e}^{E\left(\lambda \right)t}=\sum _{n\ge 0}{E}_{n}\left(\lambda \right)\frac{{t}^{n}}{n!}$
(1.10)

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

$2={e}^{E\left(\lambda \right)t}\left(\lambda {e}^{t}+1\right)=\lambda {e}^{\left(E\left(\lambda \right)+1\right)t}+{e}^{E\left(\lambda \right)t}=\sum _{n\ge 0}\left(\lambda {\left(E\left(\lambda \right)+1\right)}^{n}+{E}_{n}\left(\lambda \right)\right)\frac{{t}^{n}}{n!}.$

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

$\lambda {\left(E\left(\lambda \right)+1\right)}^{n}+{E}_{n}\left(\lambda \right)=2{\delta }_{n,0}.$
(1.11)

As is well known, the Bernoulli polynomial (see ) is also defined by the generating function to be

$\frac{t}{{e}^{t}-1}{e}^{xt}=\sum _{k\ge 0}{B}_{k}\left(x\right)\frac{{t}^{k}}{k!}.$
(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

${B}_{n}\left(x\right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){B}_{n-k}{x}^{k}.$
(1.13)

From (1.12), we note that

$\frac{t}{{e}^{t}-1}={e}^{Bt}=\sum _{n\ge 0}{B}_{n}\frac{{t}^{n}}{n!}$
(1.14)

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

$t={e}^{Bt}\left({e}^{t}-1\right)={e}^{\left(B+1\right)t}-{e}^{Bt}=\sum _{n\ge 0}\left({\left(B+1\right)}^{n}-{B}_{n}\right)\frac{{t}^{n}}{n!},$

which implies

${B}_{n}\left(1\right)-{B}_{n}={\left(B+1\right)}^{n}-{B}_{n}={\delta }_{n,1},\phantom{\rule{2em}{0ex}}{B}_{0}=1.$
(1.15)

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

$\frac{2}{{e}^{t}+1}{e}^{xt}=\sum _{k\ge 0}{E}_{k}\left(x\right)\frac{{t}^{k}}{k!}.$
(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

$\frac{2}{{e}^{t}+1}={e}^{Et}=\sum _{n\ge 0}{E}_{n}\frac{{t}^{n}}{n!}$
(1.17)

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

$2={e}^{Et}\left({e}^{t}+1\right)={e}^{\left(E+1\right)t}+{e}^{Et}=\sum _{n\ge 0}\left({\left(E+1\right)}^{n}+{E}_{n}\right)\frac{{t}^{n}}{n!},$

which implies

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

For $\lambda \in \mathbb{C}$ with $\lambda \ne -1$, the Frobenius-Euler (see [11, 1619]) polynomials are defined by

$\frac{1+\lambda }{{e}^{t}+\lambda }{e}^{xt}=\sum _{k\ge 0}{F}_{k}\left(x|-\lambda \right)\frac{{t}^{k}}{k!}.$
(1.19)

In the special case, $x=0$, ${F}_{n}\left(0|-\lambda \right)={F}_{n}\left(-\lambda \right)$ is called the nth Frobenius-Euler number (see ). By (1.19), we get

$\frac{1+\lambda }{{e}^{t}+\lambda }={e}^{Ft}=\sum _{n\ge 0}{F}_{n}\left(-\lambda \right)\frac{{t}^{n}}{n!}$
(1.20)

with the usual convention about replacing ${F}^{n}\left(-\lambda \right)$ by ${F}_{n}\left(-\lambda \right)$ (see ). By (1.19) and (1.20), we get

$1+\lambda ={e}^{F\left(-\lambda \right)t}\left({e}^{t}+\lambda \right)={e}^{\left(F\left(-\lambda \right)+1\right)t}+\lambda {e}^{F\left(-\lambda \right)t}=\sum _{n\ge 0}\left({\left(F\left(-\lambda \right)+1\right)}^{n}+\lambda {F}_{n}\left(-\lambda \right)\right)\frac{{t}^{n}}{n!},$

which implies

$\lambda {F}_{n}\left(-\lambda \right)+{F}_{n}\left(1|-\lambda \right)=\lambda {F}_{n}\left(-\lambda \right)+{\left(F\left(-\lambda \right)+1\right)}^{n}=\left(1+\lambda \right){\delta }_{n,0}.$
(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 ${\mathrm{\Pi }}_{n}$ 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 ${E}_{0}\left(x|\lambda \right),{E}_{1}\left(x|\lambda \right),\dots ,{E}_{n}\left(x|\lambda \right)$ is a good basis for ${\mathrm{\Pi }}_{n}$. Thus, for $p\left(x\right)\in {\mathrm{\Pi }}_{n}$, there exist constants ${c}_{0},{c}_{1},\dots ,{c}_{n}$ such that $p\left(x\right)={\sum }_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\lambda \right)$. Since ${E}_{n}\left(x|\lambda \right)\sim \left(\left(1+\lambda {e}^{t}\right)/2,t\right)$ (see (1.7) and (1.8)), we have

$〈\frac{1+\lambda {e}^{t}}{2}{t}^{k}|{E}_{n}\left(x|\lambda \right)〉=n!{\delta }_{n,k},$

which gives

$〈\frac{1+\lambda {e}^{t}}{2}{t}^{k}|p\left(x\right)〉=\sum _{\ell =0}^{n}{c}_{\ell }〈\frac{1+\lambda {e}^{t}}{2}{t}^{k}|{E}_{\ell }\left(x|\lambda \right)〉=\sum _{\ell =0}^{n}{c}_{\ell }\ell !{\delta }_{\ell ,k}=k!{c}_{k}.$

Hence, we can state the following result.

Theorem 2.1 For all $p\left(x\right)\in {\mathrm{\Pi }}_{n}$, there exist constants ${c}_{0},{c}_{1},\dots ,{c}_{n}$ such that $p\left(x\right)={\sum }_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\lambda \right)$, where

${c}_{k}=\frac{1}{2k!}〈\left(1+\lambda {e}^{t}\right){t}^{k}|p\left(x\right)〉.$

Now, we present several applications for our theorem. As a first application, let us take $p\left(x\right)={x}^{n}$ with $n\ge 0$. By Theorem 2.1, we have ${x}^{n}={\sum }_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\lambda \right)$, where

${c}_{k}=\frac{1}{2k!}〈\left(1+\lambda {e}^{t}\right){t}^{k}|{x}^{n}〉=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)〈1+\lambda {e}^{t}|{x}^{n-k}〉=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({\delta }_{n-k,0}+\lambda \right),$

which implies the following identity.

Corollary 2.2 For all $n\ge 0$,

${x}^{n}=\frac{1}{2}{E}_{n}\left(x|\lambda \right)+\frac{\lambda }{2}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{k}\left(x|\lambda \right).$

Let $p\left(x\right)={B}_{n}\left(x\right)\in {\mathrm{\Pi }}_{n}$, then by Theorem 2.1 we have that ${B}_{n}\left(x\right)={\sum }_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\lambda \right)$, where

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}〈\left(1+\lambda {e}^{t}\right){t}^{k}|{B}_{n}\left(x\right)〉=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)〈1+\lambda {e}^{t}|{B}_{n-k}\left(x\right)〉\\ =\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({B}_{n-k}+\lambda {B}_{n-k}\left(1\right)\right),\end{array}$

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

Corollary 2.3 For all $n\ge 2$,

${B}_{n}\left(x\right)=\frac{\left(\lambda -1\right)n}{4}{E}_{n-1}\left(x|\lambda \right)+\frac{1+\lambda }{2}\sum _{k=0,k\ne n-1}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){B}_{n-k}{E}_{k}\left(x|\lambda \right).$

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

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}〈\left(1+\lambda {e}^{t}\right){t}^{k}|{E}_{n}\left(x\right)〉=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)〈1+\lambda {e}^{t}|{E}_{n-k}\left(x\right)〉\\ =\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({E}_{n-k}+\lambda {E}_{n-k}\left(1\right)\right),\end{array}$

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

Corollary 2.4 For all $n\ge 0$,

${E}_{n}\left(x\right)=\frac{1+\lambda }{2}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{n-k}{E}_{k}\left(x|\lambda \right).$

For another application, let $p\left(x\right)={F}_{n}\left(x|-\lambda \right)$, then by Theorem 2.1 we have that ${F}_{n}\left(x|-\lambda \right)={\sum }_{k=0}^{n}{c}_{k}{E}_{k}\left(x|\lambda \right)$, where

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}〈\left(1+\lambda {e}^{t}\right){t}^{k}|{F}_{n}\left(x|-\lambda \right)〉=\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)〈1+\lambda {e}^{t}|{F}_{n-k}\left(x|-\lambda \right)〉\\ =\frac{1}{2}\left(\genfrac{}{}{0}{}{n}{k}\right)\left({F}_{n-k}\left(-\lambda \right)+\lambda {F}_{n-k}\left(1|-\lambda \right)\right),\end{array}$

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

Corollary 2.5 For all $n\ge 1$,

${F}_{n}\left(x|-\lambda \right)=\frac{1+\lambda }{2}{E}_{n}\left(x|\lambda \right)+\frac{1-{\lambda }^{2}}{2}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0}{}{n}{k}\right){F}_{n-k}\left(-\lambda \right){E}_{k}\left(x|\lambda \right).$

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

$\begin{array}{rl}{c}_{k}& =\frac{1}{2k!}\sum _{\ell =0}^{n}\frac{\left(n+\ell \right)!}{\left(n-\ell \right)!\ell !{2}^{\ell }}〈1+\lambda {e}^{t}|{t}^{k}{x}^{\ell }〉\\ =\frac{1}{2}\sum _{\ell =k}^{n}\frac{\left(n+\ell \right)!}{\left(n-\ell \right)!\ell !{2}^{\ell }}\left(\genfrac{}{}{0}{}{\ell }{k}\right)〈1+\lambda {e}^{t}|{x}^{\ell -k}〉\\ =\frac{1}{2}\sum _{\ell =k}^{n}\frac{\left(n+\ell \right)!}{\left(n-\ell \right)!\ell !{2}^{\ell }}\left(\genfrac{}{}{0}{}{\ell }{k}\right)\left({\delta }_{n-k,0}+\lambda \right)\\ =\frac{k!}{{2}^{k+1}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+k}{k}\right)+\lambda \sum _{\ell =k}^{n}\frac{k!}{{2}^{\ell +1}}\left(\genfrac{}{}{0}{}{\ell }{k}\right)\left(\genfrac{}{}{0}{}{n}{\ell }\right)\left(\genfrac{}{}{0}{}{n+\ell }{\ell }\right),\end{array}$

which implies the following identity.

Corollary 2.6 For all $n\ge 1$,

${y}_{n}\left(x\right)=\sum _{k=0}^{n}\frac{k!}{{2}^{k+1}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{n+k}{k}\right){E}_{k}\left(x|\lambda \right)+\lambda \sum _{k=0}^{n}\sum _{\ell =k}^{n}\frac{k!}{{2}^{\ell +1}}\left(\genfrac{}{}{0}{}{\ell }{k}\right)\left(\genfrac{}{}{0}{}{n}{\ell }\right)\left(\genfrac{}{}{0}{}{n+\ell }{\ell }\right){E}_{k}\left(x|\lambda \right).$

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

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

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

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• DOI: https://doi.org/10.1186/1687-1847-2013-301

### Keywords

• Bernoulli polynomial
• Bessel polynomial
• Euler polynomial
• Frobenius-Euler polynomial
• umbral calculus 