Theory and Modern Applications

# Frobenius-Euler polynomials and umbral calculus in the p-adic case

## Abstract

In this paper, we study some p-adic Frobenius-Euler measure related to umbral calculus in the p-adic case. Finally, we derive some identities of Frobenius-Euler polynomials from our study.

MSC:05A10, 05A19.

## 1 Introduction

Let p be a fixed prime number. Throughout this paper ${\mathbb{Z}}_{p}$, ${\mathbb{Q}}_{p}$ and ${\mathbb{C}}_{p}$ will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of ${\mathbb{Q}}_{p}$, respectively.

For $f\in \mathbb{N}$ with $\left(f,p\right)=1$, let

Note that the natural map induces

$\pi :X\to {\mathbb{Z}}_{p}.$

If g is a function on ${\mathbb{Z}}_{p}$, we denote by the same g the function $g\circ \pi$ on X. Namely, we can consider g as a function on X.

For $k\ge 0$ and $\lambda \in {\mathbb{C}}_{p}$ with ${|1-\lambda |}_{p}>1$, the Frobenius-Euler measure on X is defined by

${\mu }_{\lambda }\left(x+f{p}^{N}{\mathbb{Z}}_{p}\right)=\frac{{\lambda }^{f{p}^{N}-x}}{1-{\lambda }^{f{p}^{N}}}\phantom{\rule{1em}{0ex}}\left(\text{see [5, 8])},$
(1.1)

where the p-adic absolute value on ${\mathbb{C}}_{p}$ is normalized by ${|p|}_{p}=\frac{1}{p}$.

As is well known, the Frobenius-Euler polynomials are defined by the generating function to be

$\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right){e}^{xt}={e}^{H\left(x|\lambda \right)t}=\sum _{n=0}^{\mathrm{\infty }}{H}_{n}\left(x|\lambda \right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(\text{see [5, 7, 9])},$
(1.2)

with the usual convention about replacing ${H}^{n}\left(x|\lambda \right)$ by ${H}_{n}\left(x|\lambda \right)$. In the special case, $x=0$, ${H}_{n}\left(0|\lambda \right)={H}_{n}\left(\lambda \right)$ are called the n th Frobenius-Euler numbers

${H}_{n}\left(x|\lambda \right)={\left(H\left(\lambda \right)+x\right)}^{n}=\sum _{l=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{n}{l}\right){H}_{l}\left(\lambda \right){x}^{n-l}\phantom{\rule{1em}{0ex}}\text{(see [6, 9, 10])}.$
(1.3)

Thus, by (1.2) and (1.3), we easily get

${\left(H\left(\lambda \right)+1\right)}^{n}-\lambda {H}_{n}\left(\lambda \right)=\left(1-\lambda \right){\delta }_{0,n}\phantom{\rule{1em}{0ex}}\left(\text{see [1–19])},$
(1.4)

where ${\delta }_{n,k}$ is the Kronecker symbol.

For $r\in \mathbb{N}$, the Frobenius-Euler polynomials of order r are defined by the generating function

$\begin{array}{rl}{\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)}^{r}{e}^{xt}& =\underset{r\text{-times}}{\underset{⏟}{\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)×\cdots ×\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)}}{e}^{xt}\\ =\sum _{n=0}^{\mathrm{\infty }}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(\text{see [5, 9])}.\end{array}$
(1.5)

In the special case, $x=0$, ${H}_{n}^{\left(r\right)}\left(0|\lambda \right)={H}_{n}^{\left(r\right)}\left(\lambda \right)$ are called the n th Frobenius-Euler numbers of order r. The n th Frobenius-Euler polynomials can be represented by (1.1) as follows:

$\begin{array}{rl}\frac{\lambda {H}_{n}\left(x|\lambda \right)}{1-\lambda }& ={\int }_{X}{\left(x+y\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)={\int }_{{\mathbb{Z}}_{p}}{\left(x+y\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)\\ =\underset{N\to \mathrm{\infty }}{lim}\frac{1}{1-{\lambda }^{{p}^{N}}}\sum _{y=0}^{{p}^{N}-1}{\left(x+y\right)}^{n}{\lambda }^{{p}^{N}-y}\phantom{\rule{1em}{0ex}}\left(\text{see [6, 7])}.\end{array}$
(1.6)

Let be the set of all formal power series in the variable t over ${\mathbb{C}}_{p}$ with

$\mathcal{F}=\left\{f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{a}_{k}}{k!}{t}^{k}|{a}_{k}\in {\mathbb{C}}_{p}\right\}.$
(1.7)

Let $\mathbb{P}={\mathbb{C}}_{p}\left[x\right]$ and ${\mathbb{P}}^{\ast }$ denote the vector space of all linear functionals on .

The formal power series

$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{a}_{k}}{k!}{t}^{k}\in \mathcal{F}\phantom{\rule{1em}{0ex}}\left(\text{see [11, 15])}$
(1.8)

defines a linear functional on by setting

(1.9)

From (1.8) and (1.9), we have

$〈{t}^{n}|{x}^{n}〉=n!{\delta }_{n,k}\phantom{\rule{1em}{0ex}}\left(n,k\ge 0\right).$
(1.10)

Here, denotes both the algebra of formal power series in t and the vector space of all linear functionals on , and so an element $f\left(t\right)$ of will be thought of as both a formal power series and a linear functional (see [11, 15]). We will call the umbral algebra. The umbral calculus is the study of umbral algebra (see [11, 15]).

The order $o\left(f\left(t\right)\right)$ of power series $f\left(t\right)$ (≠0) is the smallest integer k for which ${a}_{k}$ does not vanish (see [11, 15]). A series $f\left(t\right)$ for which $o\left(f\left(t\right)\right)=1$ is called a delta series. If a series $f\left(t\right)$ has $o\left(f\left(t\right)\right)=0$, then $f\left(t\right)$ is called an invertible series (see [11, 15]). Let $f\left(t\right),g\left(t\right)\in \mathcal{F}$. Then we easily see that $〈f\left(t\right)g\left(t\right)|p\left(x\right)〉=〈f\left(t\right)|g\left(t\right)p\left(x\right)〉=〈g\left(t\right)|f\left(t\right)p\left(x\right)〉$. From (1.10), we note that

$〈{e}^{yt}|{x}^{n}〉={y}^{n},\phantom{\rule{2em}{0ex}}〈{e}^{yt}|p\left(x\right)〉=p\left(y\right),$
(1.11)
$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈f\left(t\right)|{x}^{k}〉}{k!}{t}^{k},\phantom{\rule{1em}{0ex}}f\left(t\right)\in \mathcal{F},$
(1.12)

and

(1.13)

For ${f}_{1}\left(t\right),{f}_{2}\left(t\right),\dots ,{f}_{m}\left(t\right)\in \mathcal{F}$, we have

$〈{f}_{1}\left(t\right){f}_{2}\left(t\right)\cdots {f}_{m}\left(t\right)|{x}^{n}〉=\sum _{{i}_{1}+\cdots +{i}_{m}=n}\left(\genfrac{}{}{0}{}{n}{{i}_{1},\dots ,{i}_{m}}\right)〈{f}_{1}\left(t\right)|{x}^{{i}_{1}}〉\cdots 〈{f}_{m}\left(t\right)|{x}^{{i}_{m}}〉.$
(1.14)

By (1.13), we get

${p}^{\left(k\right)}\left(x\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}=\sum _{l=k}^{n}〈{t}^{l}|p\left(x\right)〉\left(\genfrac{}{}{0}{}{l}{k}\right)\frac{k!}{l!}{x}^{l-k}$
(1.15)

and

${p}^{\left(k\right)}\left(0\right)=〈{t}^{k}|p\left(x\right)〉=〈1|{p}^{\left(k\right)}\left(x\right)〉.$

Thus, by (1.15), we get

${t}^{k}p\left(x\right)={p}^{\left(k\right)}\left(x\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}\phantom{\rule{1em}{0ex}}\text{(see [11, 15])}.$
(1.16)

By (1.16), we easily see that

${e}^{yt}p\left(x\right)=p\left(x+y\right)\phantom{\rule{1em}{0ex}}\text{(see [15])}.$
(1.17)

Let ${S}_{n}\left(x\right)$ denote a polynomial of degree n. Suppose that $f\left(t\right),g\left(t\right)\in \mathcal{F}$ with $o\left(f\left(t\right)\right)=1$ and $o\left(g\left(t\right)\right)=0$. Then there exists a unique sequence ${S}_{n}\left(x\right)$ of polynomials satisfying $〈g\left(t\right)f{\left(t\right)}^{k}|{S}_{n}\left(x\right)〉=n!{\delta }_{n,k}$ 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 ${S}_{n}\left(x\right)\sim \left(g\left(t\right),f\left(t\right)\right)$. If ${S}_{n}\left(x\right)\sim \left(g\left(t\right),t\right)$, then ${S}_{n}\left(x\right)$ is called the Appell sequence for $g\left(t\right)$ (see [15]).

For $p\left(x\right)\in \mathbb{P}$, we have

$〈f\left(t\right)|xp\left(x\right)〉=〈{\partial }_{t}f\left(t\right)|p\left(x\right)〉=〈{f}^{\mathrm{\prime }}\left(t\right)|p\left(x\right)〉,$
(1.18)
$〈{e}^{yt}-1|p\left(x\right)〉=p\left(y\right)-p\left(0\right).$
(1.19)

If ${S}_{n}\left(x\right)\sim \left(g\left(t\right),f\left(t\right)\right)$, then we have

$h\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈h\left(t\right)|{S}_{k}\left(x\right)〉}{k!}g\left(t\right)f{\left(t\right)}^{k},\phantom{\rule{1em}{0ex}}h\left(t\right)\in \mathcal{F},$
(1.20)
$p\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈g\left(t\right)f{\left(t\right)}^{k}|p\left(x\right)〉}{k!}{S}_{k}\left(x\right),\phantom{\rule{1em}{0ex}}p\left(x\right)\in \mathbb{P},$
(1.21)
$f\left(t\right){S}_{n}\left(x\right)=n{S}_{n-1}\left(x\right),$
(1.22)

and

(1.23)

where $\overline{f}\left(t\right)$ is compositional inverse of $f\left(t\right)$ (see [11, 15]). In [9], Kim and Kim have studied some identities of Frobenius-Euler polynomials arising from umbral calculus. In this paper, we study some p-adic Frobenius-Euler integral on related to umbral calculus in the p-adic case. Finally, we derive some new and interesting identities of Frobenius-Euler polynomials from our study.

## 2 Frobenius-Euler polynomials associated with umbral calculus

Let

$g\left(t;\lambda \right)=\frac{{e}^{t}-\lambda }{1-\lambda }\in \mathcal{F}.$
(2.1)

Then we see that $g\left(t;\lambda \right)$ is an invertible series. From (1.2), we have

$\sum _{k=0}^{\mathrm{\infty }}{H}_{k}\left(x|\lambda \right)\frac{{t}^{k}}{k!}=\frac{1}{g\left(t;\lambda \right)}{e}^{xt}.$
(2.2)

Hence, by (2.2), we get

$\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right){x}^{n}=\frac{1}{g\left(t;\lambda \right)}{x}^{n}={H}_{n}\left(x|\lambda \right).$
(2.3)

By (2.2) and (2.3), we get

${H}_{n}\left(x|\lambda \right)\sim \left(g\left(t;\lambda \right),t\right).$

From (1.6), we have

${\int }_{{\mathbb{Z}}_{p}}{e}^{\left(x+y\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)=\frac{\lambda }{{e}^{t}-\lambda }{e}^{xt},$
(2.4)

and

${\int }_{{\mathbb{Z}}_{p}}{e}^{\left(x+y+1\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)-\lambda {\int }_{{\mathbb{Z}}_{p}}{e}^{\left(x+y\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)=\lambda {e}^{xt}.$
(2.5)

By (2.5), we get

${\int }_{{\mathbb{Z}}_{p}}{\left(x+y+1\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)-\lambda {\int }_{{\mathbb{Z}}_{p}}{\left(x+y\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)=\lambda {x}^{n}.$
(2.6)

From (1.6) and (2.6), we have

$\frac{\lambda }{1-\lambda }{H}_{n}\left(x+1|\lambda \right)-\frac{{\lambda }^{2}}{1-\lambda }{H}_{n}\left(x|\lambda \right)=\lambda {x}^{n}.$
(2.7)

From (2.2), we can easily derive

${H}_{n+1}\left(x|\lambda \right)=\left(x-\frac{{g}^{\prime }\left(t;\lambda \right)}{g\left(t;\lambda \right)}\right){H}_{n}\left(x|\lambda \right).$
(2.8)

By (2.8), we get

$g\left(t;\lambda \right){H}_{n+1}\left(x|\lambda \right)=g\left(t;\lambda \right)x{H}_{n}\left(x|\lambda \right)-{g}^{\prime }\left(t;\lambda \right){H}_{n}\left(x|\lambda \right).$
(2.9)

Thus, from (2.9), we have

$\left({e}^{t}-\lambda \right){H}_{n+1}\left(x|\lambda \right)=\left({e}^{t}-\lambda \right)x{H}_{n}\left(x|\lambda \right)-{e}^{t}{H}_{n}\left(x|\lambda \right).$
(2.10)

By (2.10), we get

${H}_{n+1}\left(x+1|\lambda \right)-\lambda {H}_{n+1}\left(x|\lambda \right)=x\left({H}_{n}\left(x+1|\lambda \right)-\lambda {H}_{n}\left(x|\lambda \right)\right).$
(2.11)

From (2.11), we note that

$\begin{array}{rl}{H}_{n}\left(x+1|\lambda \right)-\lambda {H}_{n}\left(x|\lambda \right)& =x\left({H}_{n-1}\left(x+1|\lambda \right)-\lambda {H}_{n-1}\left(x|\lambda \right)\right)\\ ={x}^{2}\left({H}_{n-2}\left(x+1|\lambda \right)-\lambda {H}_{n-2}\left(x|\lambda \right)\right)=\cdots \\ ={x}^{n}\left({H}_{0}\left(x+1|\lambda \right)-\lambda {H}_{0}\left(x|\lambda \right)\right)={x}^{n}\left(1-\lambda \right).\end{array}$
(2.12)

Let us consider the linear functional $f\left(t\right)$ such that

$〈f\left(t\right)|p\left(x\right)〉={\int }_{{\mathbb{Z}}_{p}}p\left(u\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(u\right)$
(2.13)

for all polynomials $p\left(x\right)$ can be determined from (1.12) to be

$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈f\left(t\right)|{x}^{k}〉}{k!}{t}^{k}=\sum _{k=0}^{\mathrm{\infty }}{\int }_{{\mathbb{Z}}_{p}}{u}^{k}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(u\right)\frac{{t}^{k}}{k!}={\int }_{{\mathbb{Z}}_{p}}{e}^{ut}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(u\right).$
(2.14)

By (2.4) and (2.14), we get

$f\left(t\right)={\int }_{{\mathbb{Z}}_{p}}{e}^{ut}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(u\right)=\frac{\lambda }{{e}^{t}-\lambda }.$
(2.15)

Therefore, by (2.15), we obtain the following theorem.

Theorem 2.1 For $p\left(x\right)\in \mathbb{P}$, we have

$〈\frac{\lambda }{{e}^{t}-\lambda }|p\left(x\right)〉={\int }_{{\mathbb{Z}}_{p}}p\left(u\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(u\right).$

In particular,

$\frac{\lambda }{1-\lambda }{H}_{n}\left(\lambda \right)=〈{\int }_{{\mathbb{Z}}_{p}}{e}^{yt}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)|{x}^{n}〉.$

From (1.6), we have

$\sum _{n=0}^{\mathrm{\infty }}{\int }_{{\mathbb{Z}}_{p}}{\left(x+y\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)\frac{{t}^{n}}{n!}={\int }_{{\mathbb{Z}}_{p}}{e}^{\left(x+y\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)=\sum _{n=0}^{\mathrm{\infty }}{\int }_{{\mathbb{Z}}_{p}}{e}^{yt}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right){x}^{n}\frac{{t}^{n}}{n!}.$
(2.16)

By (1.6), (2.4) and (2.16), we get

(2.17)

Therefore, by (2.17), we obtain the following theorem.

Theorem 2.2 For $p\left(x\right)\in \mathbb{P}$, we have

${\int }_{{\mathbb{Z}}_{p}}p\left(x+y\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)={\int }_{{\mathbb{Z}}_{p}}{e}^{yt}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right)p\left(x\right)=\frac{\lambda }{{e}^{t}-\lambda }p\left(x\right).$

In particular,

$\frac{\lambda }{1-\lambda }{H}_{n}\left(x|\lambda \right)={\int }_{{\mathbb{Z}}_{p}}{e}^{yt}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left(y\right){x}^{n}=\frac{\lambda }{{e}^{t}-\lambda }{x}^{n}\phantom{\rule{1em}{0ex}}\left(n\ge 0\right).$

By (1.6) and (2.16), we get

$\frac{\lambda }{1-\lambda }{H}_{n}\left(x|\lambda \right)\sim \left(\frac{{e}^{t}-\lambda }{\lambda },t\right).$
(2.18)

From Appell identity and (2.18), we can derive the following identities:

${H}_{n}\left(x+y|\lambda \right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){H}_{k}\left(x|\lambda \right){y}^{n-k}.$
(2.19)

Let

${g}^{r}\left(t;\lambda \right)={\left(\frac{{e}^{t}-\lambda }{\lambda }\right)}^{r}=\underset{r\text{-times}}{\underset{⏟}{\left(\frac{{e}^{t}-\lambda }{\lambda }\right)×\cdots ×\left(\frac{{e}^{t}-\lambda }{\lambda }\right)}}\in \mathcal{F}.$
(2.20)

Then ${g}^{r}\left(t;\lambda \right)$ is an invertible functional in . By (1.5) and (2.20), we get

$\frac{1}{{g}^{r}\left(t;\lambda \right)}{e}^{xt}=\frac{{\lambda }^{r}}{{\left(1-\lambda \right)}^{r}}\sum _{k=0}^{\mathrm{\infty }}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\frac{{t}^{n}}{n!}.$
(2.21)

Thus, from (2.21), we have

$\frac{1}{{g}^{r}\left(t;\lambda \right)}{x}^{n}={\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right),$
(2.22)

and

${\left(\frac{\lambda }{1-\lambda }\right)}^{r}t{H}_{n}^{\left(r\right)}\left(x|\lambda \right)=\frac{n}{{g}^{r}\left(t;\lambda \right)}{x}^{n-1}=n{\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n-1}^{\left(r\right)}\left(x|\lambda \right).$
(2.23)

By (2.22) and (2.23), we see that

${\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\sim \left({g}^{r}\left(t;\lambda \right),t\right).$
(2.24)

From (2.4), we can derive the following identity:

$\begin{array}{r}\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}+x\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)\\ \phantom{\rule{1em}{0ex}}={\left(\frac{\lambda }{{e}^{t}-\lambda }\right)}^{r}{e}^{xt}={\left(\frac{\lambda }{1-\lambda }\right)}^{r}\sum _{n=0}^{\mathrm{\infty }}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\frac{{t}^{n}}{n!}.\end{array}$
(2.25)

By (1.10) and (2.25), we get

$\begin{array}{r}{\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\\ \phantom{\rule{1em}{0ex}}=〈\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}+x\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)|{x}^{n}〉.\end{array}$
(2.26)

From (1.14), we have

$\begin{array}{r}〈\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)|{x}^{n}〉\\ \phantom{\rule{1em}{0ex}}=\sum _{n={i}_{1}+\cdots +{i}_{r}}\left(\genfrac{}{}{0}{}{n}{{i}_{1},\dots ,{i}_{r}}\right)〈{\int }_{{\mathbb{Z}}_{p}}{e}^{{x}_{1}t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)|{x}^{{i}_{1}}〉×\cdots \\ \phantom{\rule{2em}{0ex}}×〈{\int }_{{\mathbb{Z}}_{p}}{e}^{{x}_{r}t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)|{x}^{{i}_{r}}〉\\ \phantom{\rule{1em}{0ex}}=\sum _{n={i}_{1}+\cdots +{i}_{r}}\left(\genfrac{}{}{0}{}{n}{{i}_{1},\dots ,{i}_{r}}\right){\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{{i}_{1}}\left(x|\lambda \right)\cdots {H}_{{i}_{r}}\left(x|\lambda \right).\end{array}$
(2.27)

By (2.26) and (2.27), we get

${H}_{n}^{\left(r\right)}\left(x|\lambda \right)=\sum _{n={i}_{1}+\cdots +{i}_{r}}\left(\genfrac{}{}{0}{}{n}{{i}_{1},\dots ,{i}_{r}}\right){H}_{{i}_{1}}\left(x|\lambda \right)\cdots {H}_{{i}_{r}}\left(x|\lambda \right),$

where $\left(\genfrac{}{}{0}{}{n}{{i}_{1},\dots ,{i}_{r}}\right)=\frac{n!}{{i}_{1}!\cdots {i}_{r}!}$. From (2.25), we note that

${g}^{r}\left(t;\lambda \right)=\frac{1}{\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)}={\left(\frac{{e}^{t}-\lambda }{\lambda }\right)}^{r}.$
(2.28)

Thus, by (2.28), we get

$\begin{array}{rl}\frac{1}{{g}^{r}\left(t;\lambda \right)}{e}^{xt}& ={\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right){e}^{xt}\\ ={\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}+x\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)\\ ={\left(\frac{\lambda }{1-\lambda }\right)}^{r}\sum _{n=0}^{\mathrm{\infty }}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\frac{{t}^{n}}{n!}.\end{array}$
(2.29)

By (2.29), we see that

$\begin{array}{r}{\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\\ \phantom{\rule{1em}{0ex}}=\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{\left(x+{x}_{1}+\cdots +{x}_{r}\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right){x}^{n}\\ \phantom{\rule{1em}{0ex}}=\frac{1}{{g}^{r}\left(t;\lambda \right)}{x}^{n}.\end{array}$
(2.30)

Therefore, by (2.30), we obtain the following theorem.

Theorem 2.3 For $p\left(x\right)\in \mathbb{P}$ and $r\in \mathbb{N}$, we have

$\underset{r\mathit{\text{-times}}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}p\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}+x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)={\left(\frac{\lambda }{{e}^{t}-\lambda }\right)}^{r}p\left(x\right).$

In particular,

${\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)={\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right){x}^{n}.$

Moreover,

${\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\sim \left(\frac{1}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)},t\right).$

Let us consider the function ${f}^{\ast }\left(t\right)$ in such that

$〈{f}^{\ast }\left(t\right)|p\left(x\right)〉={\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}p\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)$
(2.31)

for all polynomials $p\left(x\right)$ can be determined from (1.12) to be

$\begin{array}{rl}{f}^{\ast }\left(t\right)& =\sum _{k=0}^{\mathrm{\infty }}\frac{〈{f}^{\ast }\left(t\right)|{x}^{k}〉}{k!}{t}^{k}\\ =\sum _{k=0}^{\mathrm{\infty }}\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{\left({x}_{1}+\cdots +{x}_{r}\right)}^{k}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)\frac{{t}^{k}}{k!}\\ =\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right).\end{array}$
(2.32)

Therefore, by (2.31) and (2.32), we obtain the following theorem.

Theorem 2.4 For $p\left(x\right)\in \mathbb{P}$, we have

$\begin{array}{r}〈\underset{r\mathit{\text{-times}}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)|p\left(x\right)〉\\ \phantom{\rule{1em}{0ex}}=\underset{r\mathit{\text{-times}}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}p\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right).\end{array}$

In particular,

$〈{\left(\frac{\lambda }{{e}^{t}-\lambda }\right)}^{r}|p\left(x\right)〉=\underset{r\mathit{\text{-times}}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}p\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right).$

Indeed, the n th Frobenius-Euler number of order r is given by

${\left(\frac{\lambda }{1-\lambda }\right)}^{r}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)=〈\underset{r\text{-times}}{\underset{⏟}{{\int }_{{\mathbb{Z}}_{p}}\cdots {\int }_{{\mathbb{Z}}_{p}}}}{e}^{\left({x}_{1}+{x}_{2}+\cdots +{x}_{r}\right)t}\phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{1}\right)\cdots \phantom{\rule{0.2em}{0ex}}d{\mu }_{\lambda }\left({x}_{r}\right)|{x}^{n}〉,$

where $n\ge 0$.

Remark From (1.2) and (1.5), we note that

$\begin{array}{rl}\frac{d}{d\lambda }\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)& =\frac{1-{e}^{t}}{{\left({e}^{t}-\lambda \right)}^{2}}=\frac{1}{\left(1-\lambda \right)}\left(\frac{{\left(1-\lambda \right)}^{2}}{{\left({e}^{t}-\lambda \right)}^{2}}-\frac{1-\lambda }{{e}^{t}-\lambda }\right)\\ =\frac{1}{1-\lambda }\sum _{n=0}^{\mathrm{\infty }}\left({H}_{n}^{\left(2\right)}\left(\lambda \right)-{H}_{n}\left(\lambda \right)\right)\frac{{t}^{n}}{n!},\end{array}$
(2.33)

and

$\begin{array}{rl}\frac{{d}^{2}}{d{\lambda }^{2}}\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)& =2!\frac{1-{e}^{t}}{{\left({e}^{t}-\lambda \right)}^{3}}=\frac{2!}{{\left(1-\lambda \right)}^{2}}\left(\frac{{\left(1-\lambda \right)}^{3}}{{\left({e}^{t}-\lambda \right)}^{3}}-\frac{{\left(1-\lambda \right)}^{2}}{{\left({e}^{t}-\lambda \right)}^{2}}\right)\\ =\frac{2!}{{\left(1-\lambda \right)}^{2}}\sum _{n=0}^{\mathrm{\infty }}\left({H}_{n}^{\left(3\right)}\left(\lambda \right)-{H}_{n}^{\left(2\right)}\left(\lambda \right)\right)\frac{{t}^{n}}{n!}.\end{array}$
(2.34)

Continuing this process, we obtain the following equation:

$\begin{array}{rl}\frac{{d}^{k}}{d{\lambda }^{k}}\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)& =\frac{k!}{{\left(1-\lambda \right)}^{k}}\left(\frac{{\left(1-\lambda \right)}^{k+1}}{{\left({e}^{t}-\lambda \right)}^{k+1}}-\frac{{\left(1-\lambda \right)}^{k}}{{\left({e}^{t}-\lambda \right)}^{k}}\right)\\ =\frac{k!}{{\left(1-\lambda \right)}^{k}}\sum _{n=0}^{\mathrm{\infty }}\left({H}_{n}^{\left(k+1\right)}\left(\lambda \right)-{H}_{n}^{\left(k\right)}\left(\lambda \right)\right)\frac{{t}^{n}}{n!}.\end{array}$
(2.35)

By (1.2), (1.5) and (2.35), we get

$\frac{{d}^{k}}{d{\lambda }^{k}}{H}_{n}\left(\lambda \right)=\frac{k!}{{\left(1-\lambda \right)}^{k}}\left({H}_{n}^{\left(k+1\right)}\left(\lambda \right)-{H}_{n}^{\left(k\right)}\left(\lambda \right)\right),$

where k is a positive integer.

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## 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.

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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, D.S., Kim, T., Lee, SH. et al. Frobenius-Euler polynomials and umbral calculus in the p-adic case. Adv Differ Equ 2012, 222 (2012). https://doi.org/10.1186/1687-1847-2012-222