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# Apostol-Euler polynomials arising from umbral calculus

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

• Accepted: 10 October 2013
• Published:

## 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.

## Keywords

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

## 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 [14]). 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 [14]). 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 [14]).

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 [14]). 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 [16]).

For $\lambda \in \mathbb{C}$ with $\lambda \ne -1$, the Apostol-Euler polynomials (see [710]) 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 [1114]) 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 [17]). 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 [17]). 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.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
Department of Mathematics, Kwangwoon University, Seoul, S. Korea
(2)
Department of Mathematics, University of Haifa, Haifa, 3498838, Israel
(3)
Department of Mathematics Education, Kyungpook National University, Taegu, S. Korea
(4)
Division of General Education, Kwangwoon University, Seoul, S. Korea

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