Theory and Modern Applications

# Some identities of Frobenius-Euler polynomials arising from umbral calculus

## Abstract

In this paper, we study some interesting identities of Frobenius-Euler polynomials arising from umbral calculus.

## 1 Introduction

Let C be the complex number field, and let F be the set of all formal power series in the variable t over C with

$\mathbf{F}=\left\{f\left(t\right)=\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{{a}_{k}}{k!}{t}^{k}|{a}_{k}âˆˆ\mathbf{C}\right\}.$

We use notation $\mathbb{P}=\mathbf{C}\left[x\right]$ and ${\mathbb{P}}^{âˆ—}$ denotes the vector space of all linear functional on â„™.

Also, $ã€ˆL|p\left(x\right)ã€‰$ denotes the action of the linear functional L on the polynomial $p\left(x\right)$, and we remind that the vector space operations on ${\mathbb{P}}^{âˆ—}$ is defined by

$\begin{array}{c}ã€ˆL+M|p\left(x\right)ã€‰=ã€ˆL|p\left(x\right)ã€‰+ã€ˆM|p\left(x\right)ã€‰,\hfill \\ ã€ˆcL|p\left(x\right)ã€‰=cã€ˆL|p\left(x\right)ã€‰\phantom{\rule{1em}{0ex}}\left(\text{see [1]}\right),\hfill \end{array}$

where c is any constant in C.

The formal power series

$f\left(t\right)=\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{{a}_{k}}{k!}{t}^{k}âˆˆ\mathbf{F}\phantom{\rule{1em}{0ex}}\left(\text{see [1, 2]}\right),$
(1)

defines a linear functional on â„™ by setting

(2)

In particular,

$ã€ˆ{t}^{k}|{x}^{n}ã€‰=n!{\mathrm{Î´}}_{n,k},$
(3)

where ${\mathrm{Î´}}_{n,k}$ is the Kronecker symbol. If ${f}_{L}\left(t\right)={âˆ‘}_{k=0}^{\mathrm{âˆž}}\frac{ã€ˆL|{x}^{k}ã€‰}{k!}{t}^{k}$, then we get $ã€ˆ{f}_{L}\left(t\right)|{x}^{n}ã€‰=ã€ˆL|{x}^{n}ã€‰$ and so as linear functionals $L={f}_{L}\left(t\right)$ (see [1, 2]).

In addition, the map $Lâ†¦{f}_{L}\left(t\right)$ is a vector space isomorphism from ${\mathbb{P}}^{âˆ—}$ onto F (see [1, 2]). Henceforth, F will denote both the algebra of formal power series in t and the vector space of all linear functionals on â„™, and so an element $f\left(t\right)$ of F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra (see [1, 2]).

Let us give an example. For y in C the evaluation functional is defined to be the power series ${e}^{yt}$. From (2), we have $ã€ˆ{e}^{yt}|{x}^{n}ã€‰={y}^{n}$ and so $ã€ˆ{e}^{yt}|p\left(x\right)ã€‰=p\left(y\right)$ (see [1, 2]). Notice that for all $f\left(t\right)$ in F,

$f\left(t\right)=\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{ã€ˆf\left(t\right)|{x}^{t}ã€‰}{k!}{t}^{k}$
(4)

and for all polynomial $p\left(x\right)$

$p\left(x\right)=\underset{kâ‰¥0}{âˆ‘}\frac{ã€ˆ{t}^{k}|p\left(x\right)ã€‰}{k!}{x}^{k}\phantom{\rule{1em}{0ex}}\left(\text{see [1, 2]}\right).$
(5)

For ${f}_{1}\left(t\right),{f}_{2}\left(t\right),â€¦,{f}_{m}\left(t\right)âˆˆ\mathbf{F}$, we have

$\begin{array}{c}ã€ˆ{f}_{1}\left(t\right){f}_{2}\left(t\right)â‹¯{f}_{m}\left(t\right)|{x}^{n}ã€‰\hfill \\ \phantom{\rule{1em}{0ex}}=âˆ‘\left(\genfrac{}{}{0}{}{n}{{i}_{1},â€¦,{i}_{m}}\right)ã€ˆ{f}_{1}\left(t\right)|{x}^{{i}_{1}}ã€‰â‹¯ã€ˆ{f}_{n}\left(t\right)|{x}^{{i}_{m}}ã€‰,\hfill \end{array}$

where the sum is over all nonnegative integers ${i}_{1},{i}_{2},â€¦,{i}_{m}$ such that ${i}_{1}+â‹¯+{i}_{m}=n$ (see [1, 2]). The order $o\left(f\left(t\right)\right)$ of the power series is the smallest integer k for which ${a}_{k}$ does not vanish. We define $o\left(f\left(t\right)\right)=\mathrm{âˆž}$ if $f\left(t\right)=0$. We see that $o\left(f\left(t\right)g\left(t\right)\right)=o\left(f\left(t\right)\right)+o\left(g\left(t\right)\right)$ and $o\left(f\left(t\right)+g\left(t\right)\right)â‰¥min\left\{o\left(f\left(t\right)\right),o\left(g\left(t\right)\right)\right\}$. The series $f\left(t\right)$ has a multiplicative inverse, denoted by $f{\left(t\right)}^{âˆ’1}$ or $\frac{1}{f\left(t\right)}$, if and only if $o\left(f\left(t\right)\right)=0$. Such series is called an invertible series. A series $f\left(t\right)$ for which $o\left(f\left(t\right)\right)=1$ is called a delta series (see [1, 2]). For $f\left(t\right),g\left(t\right)âˆˆ\mathbf{F}$, we have $ã€ˆf\left(t\right)g\left(t\right)|p\left(x\right)ã€‰=ã€ˆf\left(t\right)|g\left(t\right)p\left(x\right)ã€‰$.

A delta series $f\left(t\right)$ has a compositional inverse $\stackrel{Â¯}{f}\left(t\right)$ such that $f\left(\stackrel{Â¯}{f}\left(t\right)\right)=\stackrel{Â¯}{f}\left(f\left(t\right)\right)=t$.

For $f\left(t\right),g\left(t\right)âˆˆ\mathbf{F}$, we have $ã€ˆf\left(t\right)g\left(t\right)|p\left(x\right)ã€‰=ã€ˆf\left(t\right)|g\left(t\right)p\left(x\right)ã€‰$.

From (5), we have

${p}^{\left(k\right)}\left(x\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}=\underset{l=k}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{ã€ˆ{t}^{l}|p\left(x\right)ã€‰}{l!}l\left(lâˆ’1\right)â‹¯\left(lâˆ’k+1\right){x}^{lâˆ’k}.$

Thus, we see that

${p}^{\left(k\right)}\left(0\right)=ã€ˆ{t}^{k}|p\left(x\right)ã€‰=ã€ˆ1|{p}^{\left(k\right)}\left(x\right)ã€‰.$
(6)

By (6), we get

${t}^{k}p\left(x\right)={p}^{\left(k\right)}\left(x\right)=\frac{{d}^{k}\left(p\left(x\right)\right)}{d{x}^{k}}\phantom{\rule{1em}{0ex}}\left(\text{see [1, 2]}\right).$
(7)

By (7), we have

${e}^{yt}p\left(x\right)=p\left(x+y\right)\phantom{\rule{1em}{0ex}}\left(\text{see [1, 2]}\right).$
(8)

Let ${S}_{n}\left(x\right)$ be a polynomial with $deg{S}_{n}\left(x\right)=n$.

Let $f\left(t\right)$ be a delta series, and let $g\left(t\right)$ be an invertible series. Then there exists a unique sequence ${S}_{n}\left(x\right)$ of polynomials such that $ã€ˆg\left(t\right)f{\left(t\right)}^{k}|{S}_{n}\left(x\right)ã€‰=n!{\mathrm{Î´}}_{n,k}$ for all $n,kâ‰¥0$. The sequence ${S}_{n}\left(x\right)$ is called the Sheffer sequence for $\left(g\left(t\right),f\left(t\right)\right)$ or that ${S}_{n}\left(t\right)$ is Sheffer for $\left(g\left(t\right),f\left(t\right)\right)$.

The Sheffer sequence for $\left(1,f\left(t\right)\right)$ is called the associated sequence for $f\left(t\right)$ or ${S}_{n}\left(x\right)$ is associated to $f\left(t\right)$. The Sheffer sequence for $\left(g\left(t\right),t\right)$ is called the Appell sequence for $g\left(t\right)$ or ${S}_{n}\left(x\right)$ is Appell for $g\left(t\right)$ (see [1, 2]). The umbral calculus is the study of umbral algebra and the modern classical umbral calculus can be described as a systemic study of the class of Sheffer sequences. Let $p\left(x\right)âˆˆ\mathbb{P}$. Then we have

(9)
(10)

and

$ã€ˆ{e}^{yt}âˆ’1|p\left(x\right)ã€‰=p\left(y\right)âˆ’p\left(0\right)\phantom{\rule{1em}{0ex}}\left(\text{see [1, 2]}\right).$
(11)

Let ${S}_{n}\left(x\right)$ be Sheffer for $\left(g\left(t\right),f\left(t\right)\right)$. Then

(12)
(13)
(14)
(15)

For , we recall that the Frobenius-Euler polynomials are defined by the generating function to be

$\frac{1âˆ’\mathrm{Î»}}{{e}^{t}âˆ’\mathrm{Î»}}{e}^{xt}={e}^{H\left(x|\mathrm{Î»}\right)t}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{H}_{n}\left(x|\mathrm{Î»}\right)\frac{{t}^{n}}{n!},$
(16)

with the usual convention about replacing ${H}^{n}\left(x|\mathrm{Î»}\right)$ by ${H}_{n}\left(x|\mathrm{Î»}\right)$ (see [3]). In the special case, $x=0$, ${H}_{n}\left(0|\mathrm{Î»}\right)={H}_{n}\left(\mathrm{Î»}\right)$ are called the n th Frobenius-Euler numbers. By (16), we get

${H}_{n}\left(x|\mathrm{Î»}\right)={\left(H\left(\mathrm{Î»}\right)+x\right)}^{n}=\underset{l=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{l}\right){H}_{nâˆ’l}^{\left(\mathrm{Î»}\right)}{x}^{l},$
(17)

and

${\left(H\left(\mathrm{Î»}\right)+1\right)}^{n}âˆ’\mathrm{Î»}{H}_{n}\left(\mathrm{Î»}\right)=\left(1âˆ’\mathrm{Î»}\right){\mathrm{Î´}}_{0,n}\phantom{\rule{1em}{0ex}}\left(\text{see [1, 4â€“13]}\right).$
(18)

From (17), we note that the leading coefficient of ${H}_{n}\left(x|\mathrm{Î»}\right)$ is ${H}_{0}\left(\mathrm{Î»}\right)=1$. So, ${H}_{n}\left(x|\mathrm{Î»}\right)$ is a monic polynomial of degree n with coefficients in $\mathbf{Q}\left(\mathrm{Î»}\right)$.

In this paper, we derive some new identities of Frobenius-Euler polynomials arising from umbral calculus.

## 2 Applications of umbral calculus to Frobenius-Euler polynomials

Let ${S}_{n}\left(x\right)$ be an Appell sequence for $g\left(t\right)$. From (14), we have

$\frac{1}{g\left(t\right)}{x}^{n}={S}_{n}\left(x\right)\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}{x}^{n}=g\left(t\right){S}_{n}\left(x\right)\phantom{\rule{1em}{0ex}}\left(nâ‰¥0\right).$
(19)

For , let us take ${g}_{\mathrm{Î»}}\left(t\right)=\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}âˆˆ\mathbf{F}$.

Then we see that ${g}_{\mathrm{Î»}}\left(t\right)$ is an invertible series.

From (16), we have

$\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{{H}_{k}\left(x|\mathrm{Î»}\right)}{k!}{t}^{k}=\frac{1}{{g}_{\mathrm{Î»}}\left(t\right)}{e}^{xt}.$
(20)

By (20), we get

(21)

and by (17), we get

$t{H}_{n}\left(x|\mathrm{Î»}\right)={H}_{n}^{\mathrm{â€²}}\left(x|\mathrm{Î»}\right)=n{H}_{nâˆ’1}\left(x|\mathrm{Î»}\right).$
(22)

Therefore, by (21) and (22), we obtain the following proposition.

Proposition 1 For , $nâ‰¥0$, we see that ${H}_{n}\left(x|\mathrm{Î»}\right)$ is the Appell sequence for ${g}_{\mathrm{Î»}}\left(t\right)=\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}$.

From (20), we have

$\begin{array}{rcl}\underset{k=1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{{H}_{k}\left(x|\mathrm{Î»}\right)}{k!}k{t}^{kâˆ’1}& =& \frac{x{g}_{\mathrm{Î»}}\left(t\right){e}^{xt}âˆ’{g}_{\mathrm{Î»}}^{\mathrm{â€²}}\left(t\right){e}^{xt}}{{g}_{\mathrm{Î»}}{\left(t\right)}^{2}}\\ =& \underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\left\{x\frac{1}{{g}_{\mathrm{Î»}}\left(t\right)}{x}^{k}âˆ’\frac{{g}_{\mathrm{Î»}}^{\mathrm{â€²}}\left(t\right)}{{g}_{\mathrm{Î»}}\left(t\right)}\frac{1}{{g}_{\mathrm{Î»}}\left(t\right)}{x}^{k}\right\}\frac{{t}^{k}}{k!}.\end{array}$
(23)

By (21) and (23), we get

${H}_{k+1}\left(x|\mathrm{Î»}\right)=x{H}_{k}\left(x|\mathrm{Î»}\right)âˆ’\frac{{g}_{\mathrm{Î»}}^{\mathrm{â€²}}\left(t\right)}{{g}_{\mathrm{Î»}}\left(t\right)}{H}_{k}\left(x|\mathrm{Î»}\right).$
(24)

Therefore, by (24) we obtain the following theorem.

Theorem 2 Let ${g}_{\mathrm{Î»}}\left(t\right)=\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}âˆˆ\mathbf{F}$. Then we have

${H}_{k+1}\left(x|\mathrm{Î»}\right)=\left(xâˆ’\frac{{g}_{\mathrm{Î»}}^{\mathrm{â€²}}\left(t\right)}{{g}_{\mathrm{Î»}}\left(t\right)}\right){H}_{k}\left(x|\mathrm{Î»}\right)\phantom{\rule{1em}{0ex}}\left(kâ‰¥0\right).$

From (16), we have

$\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\left({H}_{n}\left(x+1|\mathrm{Î»}\right)âˆ’\mathrm{Î»}{H}_{n}\left(x|\mathrm{Î»}\right)\right)\frac{{t}^{n}}{n!}=\frac{1âˆ’\mathrm{Î»}}{{e}^{t}âˆ’\mathrm{Î»}}{e}^{\left(x+1\right)t}âˆ’\mathrm{Î»}\frac{1âˆ’\mathrm{Î»}}{{e}^{t}âˆ’\mathrm{Î»}}{e}^{xt}=\left(1âˆ’\mathrm{Î»}\right){e}^{xt}.$
(25)

By (25), we get

${H}_{n}\left(x+1|\mathrm{Î»}\right)âˆ’\mathrm{Î»}{H}_{n}\left(x|\mathrm{Î»}\right)=\left(1âˆ’\mathrm{Î»}\right){x}^{n}.$
(26)

From Theorem 2, we can derive the following equation (27):

${g}_{\mathrm{Î»}}\left(t\right){H}_{k+1}\left(x|\mathrm{Î»}\right)=\left({g}_{\mathrm{Î»}}\left(t\right)xâˆ’{g}_{\mathrm{Î»}}^{\mathrm{â€²}}\left(t\right)\right){H}_{k}\left(x|\mathrm{Î»}\right).$
(27)

By (27), we get

$\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right){H}_{k+1}\left(x|\mathrm{Î»}\right)=\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}x{H}_{k}\left(x|\mathrm{Î»}\right)âˆ’\frac{{e}^{t}}{1âˆ’\mathrm{Î»}}{H}_{k}\left(x|\mathrm{Î»}\right).$
(28)

From (8) and (28), we have

$\begin{array}{rcl}{H}_{k+1}\left(x+1|\mathrm{Î»}\right)âˆ’\mathrm{Î»}{H}_{k+1}\left(x|\mathrm{Î»}\right)& =& \left(x+1\right){H}_{k}\left(x+1|\mathrm{Î»}\right)âˆ’\mathrm{Î»}x{H}_{k}\left(x|\mathrm{Î»}\right)âˆ’{H}_{k}\left(x+1|\mathrm{Î»}\right)\\ =& x{H}_{k}\left(x+1|\mathrm{Î»}\right)âˆ’\mathrm{Î»}x{H}_{k}\left(x|\mathrm{Î»}\right).\end{array}$

Therefore, by (26), we obtain the following theorem.

Theorem 3 For $kâ‰¥0$, we have

${H}_{k+1}\left(x+1|\mathrm{Î»}\right)=\mathrm{Î»}{H}_{k+1}\left(x|\mathrm{Î»}\right)+\left(1âˆ’\mathrm{Î»}\right){x}^{k+1}.$

From (16), (17), and (18), we note that

$\begin{array}{rcl}{âˆ«}_{x}^{x+y}{H}_{n}\left(u|\mathrm{Î»}\right)\phantom{\rule{0.2em}{0ex}}du& =& \frac{1}{n+1}\left\{{H}_{n+1}\left(x+y|\mathrm{Î»}\right)âˆ’{H}_{n+1}\left(x|\mathrm{Î»}\right)\right\}\\ =& \frac{1}{n+1}\underset{k=1}{\overset{\mathrm{âˆž}}{âˆ‘}}\left(\genfrac{}{}{0}{}{n+1}{k}\right){H}_{n+1âˆ’k}\left(x|\mathrm{Î»}\right){y}^{k}\\ =& \underset{k=1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{n\left(nâˆ’1\right)â‹¯\left(nâˆ’k+2\right)}{k!}{H}_{n+1âˆ’k}\left(x|\mathrm{Î»}\right){y}^{k}\\ =& \underset{k=1}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{{y}^{k}}{k!}{t}^{kâˆ’1}{H}_{n}\left(x|\mathrm{Î»}\right)\\ =& \frac{1}{t}\left(\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{{y}^{k}}{k!}{t}^{k}âˆ’1\right){H}_{n}\left(x|\mathrm{Î»}\right)\\ =& \frac{{e}^{yt}âˆ’1}{t}{H}_{n}\left(x|\mathrm{Î»}\right).\end{array}$
(29)

Therefore, by (29), we obtain the following theorem.

Theorem 4 For , $nâ‰¥0$, we have

${âˆ«}_{x}^{x+y}{H}_{n}\left(u|\mathrm{Î»}\right)\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{yt}âˆ’1}{t}{H}_{n}\left(x|\mathrm{Î»}\right).$

By (15) and Proposition 1, we get

$t\left\{\frac{1}{n+1}{H}_{n+1}\left(x|\mathrm{Î»}\right)\right\}={H}_{n}\left(x|\mathrm{Î»}\right).$
(30)

From (30), we can derive equation (31):

$\begin{array}{rcl}ã€ˆ{e}^{yt}âˆ’1|\frac{{H}_{n+1}\left(x|\mathrm{Î»}\right)}{n+1}ã€‰& =& ã€ˆ\frac{{e}^{yt}âˆ’1}{t}|t\left\{\frac{{H}_{n+1}\left(x|\mathrm{Î»}\right)}{n+1}\right\}ã€‰\\ =& ã€ˆ\frac{{e}^{yt}âˆ’1}{t}|{H}_{n}\left(x|\mathrm{Î»}\right)ã€‰.\end{array}$
(31)

By (11) and (31), we get

$\begin{array}{rcl}ã€ˆ\frac{{e}^{yt}âˆ’1}{t}|{H}_{n}\left(x|\mathrm{Î»}\right)ã€‰& =& ã€ˆ{e}^{yt}âˆ’1|\frac{{H}_{n+1}\left(x|\mathrm{Î»}\right)}{n+1}ã€‰\\ =& \frac{1}{n+1}\left\{{H}_{n+1}\left(y|\mathrm{Î»}\right)âˆ’{H}_{n+1}\left(\mathrm{Î»}\right)\right\}={âˆ«}_{0}^{y}{H}_{n}\left(u|\mathrm{Î»}\right)\phantom{\rule{0.2em}{0ex}}du.\end{array}$
(32)

Therefore, by (32), we obtain the following corollary.

Corollary 5 For $nâ‰¥0$, we have

$ã€ˆ\frac{{e}^{yt}âˆ’1}{t}|{H}_{n}\left(x|\mathrm{Î»}\right)ã€‰={âˆ«}_{0}^{y}{H}_{n}\left(u|\mathrm{Î»}\right)\phantom{\rule{0.2em}{0ex}}du.$

Let $\mathbb{P}\left(\mathrm{Î»}\right)=\left\{p\left(x\right)âˆˆ\mathbf{Q}\left(\mathrm{Î»}\right)\left[x\right]|degp\left(x\right)â‰¤n\right\}$ be a vector space over $\mathbf{Q}\left(\mathrm{Î»}\right)$.

For $p\left(x\right)âˆˆ{\mathbb{P}}_{n}\left(\mathrm{Î»}\right)$, let us take

$p\left(x\right)=\underset{k=0}{\overset{n}{âˆ‘}}{b}_{k}{H}_{k}\left(x|\mathrm{Î»}\right).$
(33)

By Proposition 1, ${H}_{n}\left(x|\mathrm{Î»}\right)$ is an Appell sequence for ${g}_{\mathrm{Î»}}\left(t\right)=\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}$ where . Thus, we have

$ã€ˆ\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}{t}^{k}|{H}_{n}\left(x|\mathrm{Î»}\right)ã€‰=n!{\mathrm{Î´}}_{n,k}.$
(34)

From (33) and (34), we can derive

$\begin{array}{rcl}ã€ˆ\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}{t}^{k}|p\left(x\right)ã€‰& =& \underset{l=0}{\overset{n}{âˆ‘}}{b}_{l}ã€ˆ\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}{t}^{k}|{H}_{l}\left(x|\mathrm{Î»}\right)ã€‰\\ =& \underset{l=0}{\overset{n}{âˆ‘}}{b}_{l}l!{\mathrm{Î´}}_{l,k}=k!{b}_{k}.\end{array}$
(35)

Thus, by (35), we get

$\begin{array}{rcl}{b}_{k}& =& \frac{1}{k!}ã€ˆ\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}{t}^{k}|p\left(x\right)ã€‰\\ =& \frac{1}{k!\left(1âˆ’\mathrm{Î»}\right)}ã€ˆ\left({e}^{t}âˆ’\mathrm{Î»}\right){t}^{k}|p\left(x\right)ã€‰\\ =& \frac{1}{k!\left(1âˆ’\mathrm{Î»}\right)}ã€ˆ{e}^{t}âˆ’\mathrm{Î»}|{p}^{\left(k\right)}\left(x\right)ã€‰.\end{array}$
(36)

From (11) and (36), we have

${b}_{k}=\frac{1}{k!\left(1âˆ’\mathrm{Î»}\right)}\left\{{p}^{\left(k\right)}\left(1\right)âˆ’\mathrm{Î»}{p}^{\left(k\right)}\left(0\right)\right\},$
(37)

where ${p}^{\left(k\right)}\left(x\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}$.

Therefore, by (37), we obtain the following theorem.

Theorem 6 For $p\left(x\right)âˆˆ{\mathbb{P}}_{n}\left(\mathrm{Î»}\right)$, let us assume that $p\left(x\right)={âˆ‘}_{k=0}^{n}{b}_{k}{H}_{k}\left(x|\mathrm{Î»}\right)$. Then we have

${b}_{k}=\frac{1}{k!\left(1âˆ’\mathrm{Î»}\right)}\left\{{p}^{\left(k\right)}\left(1\right)âˆ’\mathrm{Î»}{p}^{\left(k\right)}\left(0\right)\right\},$

where ${p}^{\left(k\right)}\left(1\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}{|}_{x=1}$.

The higher-order Frobenius-Euler polynomials are defined by

${\left(\frac{1âˆ’\mathrm{Î»}}{{e}^{t}âˆ’\mathrm{Î»}}\right)}^{r}{e}^{xt}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)\frac{{t}^{n}}{n!},$
(38)

where and $râˆˆ\mathbf{N}$ (see [4, 11]).

In the special case, $x=0$, ${H}_{n}^{\left(r\right)}\left(0|\mathrm{Î»}\right)={H}_{n}^{\left(r\right)}\left(\mathrm{Î»}\right)$ are called the n th Frobenius-Euler numbers of order r. From (38), we have

$\begin{array}{rcl}{H}_{n}^{\left(r\right)}\left(x\right)& =& \underset{l=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{l}\right){H}_{nâˆ’l}^{\left(r\right)}\left(\mathrm{Î»}\right){x}^{l}\\ =& \underset{{n}_{1}+â‹¯+{n}_{r}=n}{âˆ‘}\left(\genfrac{}{}{0}{}{n}{{n}_{1},â€¦,{n}_{r}}\right){H}_{{n}_{1}}\left(x|\mathrm{Î»}\right)â‹¯{H}_{{n}_{r}}\left(x|\mathrm{Î»}\right).\end{array}$
(39)

Note that ${H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)$ is a monic polynomial of degree n with coefficients in $\mathbf{Q}\left(\mathrm{Î»}\right)$.

For $râˆˆ\mathbf{N}$, , let ${g}_{\mathrm{Î»}}^{r}\left(t\right)={\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right)}^{r}$. Then we easily see that ${g}_{\mathrm{Î»}}^{r}\left(t\right)$ is an invertible series.

From (38) and (39), we have

$\frac{1}{{g}_{\mathrm{Î»}}^{r}\left(t\right)}{e}^{xt}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)\frac{{t}^{n}}{n!},$
(40)

and

$t{H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)=n{H}_{nâˆ’1}^{\left(r\right)}\left(x|\mathrm{Î»}\right).$
(41)

By (40), we get

$\frac{1}{{g}_{\mathrm{Î»}}^{r}\left(t\right)}{x}^{n}={H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)\phantom{\rule{1em}{0ex}}\left(nâˆˆ{\mathbf{Z}}_{+},râˆˆ\mathbf{N}\right).$
(42)

Therefore, by (41) and (42), we obtain the following proposition.

Proposition 7 For $nâˆˆ{\mathbf{Z}}_{+}$, ${H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)$ is an Appell sequence for

${g}_{\mathrm{Î»}}^{r}\left(t\right)={\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right)}^{r}.$

Moreover,

$\frac{1}{{g}_{\mathrm{Î»}}^{r}\left(t\right)}{x}^{n}={H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}t{H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)=n{H}_{nâˆ’1}^{\left(r\right)}\left(x|\mathrm{Î»}\right).$

Remark Note that

$ã€ˆ\frac{1âˆ’\mathrm{Î»}}{{e}^{t}âˆ’\mathrm{Î»}}|{x}^{n}ã€‰={H}_{n}\left(\mathrm{Î»}\right).$
(43)

From (43), we have

(44)
(45)

By (43), (44), and (45), we get

$\underset{n={i}_{1}+â‹¯+{i}_{r}}{âˆ‘}\left(\genfrac{}{}{0}{}{n}{{i}_{1},â€¦,{i}_{r}}\right){H}_{{i}_{1}}\left(\mathrm{Î»}\right)â‹¯{H}_{{i}_{r}}\left(\mathrm{Î»}\right)={H}_{n}^{\left(r\right)}\left(\mathrm{Î»}\right).$

Let us take $p\left(x\right)âˆˆ{\mathbb{P}}_{n}\left(\mathrm{Î»}\right)$ with

$p\left(x\right)=\underset{k=0}{\overset{n}{âˆ‘}}{C}_{k}^{\left(r\right)}{H}_{k}^{\left(r\right)}\left(x|\mathrm{Î»}\right).$
(46)

From the definition of Appell sequences, we have

$ã€ˆ{\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right)}^{r}|{H}_{n}^{\left(r\right)}\left(x|\mathrm{Î»}\right)ã€‰=n!{\mathrm{Î´}}_{n,k}.$
(47)

By (46) and (47), we get

$\begin{array}{rcl}ã€ˆ{\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right)}^{r}{t}^{k}|p\left(x\right)ã€‰& =& \underset{l=0}{\overset{n}{âˆ‘}}{C}_{l}^{\left(r\right)}ã€ˆ{\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right)}^{r}{t}^{k}|{H}_{l}\left(x|\mathrm{Î»}\right)ã€‰\\ =& \underset{l=0}{\overset{n}{âˆ‘}}{C}_{l}^{\left(r\right)}l!{\mathrm{Î´}}_{l,k}=k!{C}_{k}^{\left(r\right)}.\end{array}$
(48)

Thus, from (48), we have

$\begin{array}{rcl}{C}_{k}^{\left(r\right)}& =& \frac{1}{k!}ã€ˆ{\left(\frac{{e}^{t}âˆ’\mathrm{Î»}}{1âˆ’\mathrm{Î»}}\right)}^{r}{t}^{k}|p\left(x\right)ã€‰\\ =& \frac{1}{k!{\left(1âˆ’\mathrm{Î»}\right)}^{r}}ã€ˆ{\left({e}^{t}âˆ’\mathrm{Î»}\right)}^{r}{t}^{k}|p\left(x\right)ã€‰\\ =& \frac{1}{k!{\left(1âˆ’\mathrm{Î»}\right)}^{r}}\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){\left(âˆ’\mathrm{Î»}\right)}^{râˆ’l}ã€ˆ{e}^{lt}|{p}^{\left(k\right)}\left(x\right)ã€‰\\ =& \frac{1}{k!{\left(1âˆ’\mathrm{Î»}\right)}^{r}}\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){\left(âˆ’\mathrm{Î»}\right)}^{râˆ’l}{p}^{\left(k\right)}\left(l\right).\end{array}$
(49)

Therefore, by (46) and (49), we obtain the following theorem.

Theorem 8 For $p\left(x\right)âˆˆ{\mathbb{P}}_{n}\left(\mathrm{Î»}\right)$, let

$p\left(x\right)=\underset{k=0}{\overset{n}{âˆ‘}}{C}_{k}^{\left(r\right)}{H}_{k}^{\left(r\right)}\left(x|\mathrm{Î»}\right).$

Then we have

${C}_{k}^{\left(r\right)}=\frac{1}{k!{\left(1âˆ’\mathrm{Î»}\right)}^{r}}\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){\left(âˆ’\mathrm{Î»}\right)}^{râˆ’l}{p}^{\left(k\right)}\left(l\right),$

where $râˆˆ\mathbf{N}$ and ${p}^{\left(k\right)}\left(l\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}{|}_{x=l}$.

Remark Let ${S}_{n}\left(x\right)$ be a Sheffer sequence for $\left(g\left(t\right),f\left(t\right)\right)$. Then Sheffer identity is given by

${S}_{n}\left(x+y\right)=\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){P}_{k}\left(y\right){S}_{nâˆ’k}\left(x\right)=\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){P}_{k}\left(x\right){S}_{nâˆ’k}\left(y\right),$
(50)

where ${P}_{k}\left(y\right)=g\left(t\right){S}_{k}\left(y\right)$ is associated to $f\left(t\right)$ (see [1, 2]).

From (21), Proposition 1, and (50), we have

$\begin{array}{rcl}{H}_{n}\left(x+y|\mathrm{Î»}\right)& =& \underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){P}_{k}\left(y\right){S}_{nâˆ’k}\left(x\right)\\ =& \underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right){H}_{nâˆ’k}\left(y|\mathrm{Î»}\right){x}^{k}.\end{array}$

By Proposition 7 and (50), we get

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

Let . Then we have

${H}_{n}\left(\mathrm{Î±}x|\mathrm{Î»}\right)={\mathrm{Î±}}^{n}\frac{{g}_{\mathrm{Î»}}\left(t\right)}{{g}_{\mathrm{Î»}}\left(\frac{t}{\mathrm{Î±}}\right)}{H}_{n}\left(x|\mathrm{Î»}\right).$

## References

1. Roman S: The Umbral Calculus. Dover, New York; 2005.

2. Dere R, Simsek Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):433â€“438.

3. Araci S, Acikgoz M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):399â€“406.

4. Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132(12):2854â€“2865. 10.1016/j.jnt.2012.05.033

5. Kim T, Choi J: A note on the product of Frobenius-Euler polynomials arising from the p -adic integral on ${\mathbf{Z}}_{p}$ . Adv. Stud. Contemp. Math. 2012, 22(2):215â€“223.

6. Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on ${\mathbf{Z}}_{p}$ . Russ. J. Math. Phys. 2009, 16(1):93â€“96. 10.1134/S1061920809010063

7. Rim S-H, Jeong J: On the modified q -Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. 2012, 22(1):93â€“98.

8. Rim S-H, Lee J: Some identities on the twisted $\left(h,q\right)$ -Geonocchi numbers and polynomials associated with q -Bernstein polynomials. Int. J. Math. Math. Sci. 2011., 2011: Article ID 482840

9. Simsek Y, Yurekli O, Kurt V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv. Stud. Contemp. Math. 2007, 15(2):187â€“194.

10. Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495â€“501.

11. Simsek Y, Bayad A, Lokesha V: q -Bernstein polynomials related to q -Frobenius-Euler polynomials, l -functions, and q -Stirling numbers. Math. Methods Appl. Sci. 2012, 35(8):877â€“884. 10.1002/mma.1580

12. Shiratani K: On the Euler numbers. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 1973, 27: 1â€“5. 10.2206/kyushumfs.27.1

13. Shiratani K, Yamamoto S: On a p -adic interpolation function for the Euler numbers and its derivatives. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 1985, 39(1):113â€“125. 10.2206/kyushumfs.39.113

## Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786.

## Author information

Authors

### Corresponding author

Correspondence to Taekyun Kim.

### Competing interests

The authors declare that they have no competing interests.

### Authorsâ€™ contributions

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

## Rights and permissions

Reprints and permissions

Kim, D.S., Kim, T. Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv Differ Equ 2012, 196 (2012). https://doi.org/10.1186/1687-1847-2012-196