Theory and Modern Applications

# Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus

## Abstract

In this paper, we derive the identities of higher-order Bernoulli, Euler and Frobenius-Euler polynomials from the orthogonality of Hermite polynomials. Finally, we give some interesting and new identities of several special polynomials arising from umbral calculus.

MSC: 05A10, 05A19.

## 1 Introduction

The Hermite polynomials are defined by the generating function to be

${e}^{2xt-{t}^{2}}={e}^{H\left(x\right)t}=\sum _{n=0}^{\mathrm{\infty }}{H}_{n}\left(x\right)\frac{{t}^{n}}{n!}$
(1.1)

with the usual convention about replacing ${H}^{n}\left(x\right)$ by ${H}_{n}\left(x\right)$ (see ). In the special case, $x=0$, ${H}_{n}\left(0\right)={H}_{n}$ are called the nth Hermite numbers. From (1.1) we have

${H}_{n}\left(x\right)={\left(H+2x\right)}^{n}=\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){H}_{n-l}{x}^{l}{2}^{l}.$
(1.2)

Thus, by (1.2), we get

$\frac{{d}^{k}}{d{x}^{k}}{H}_{n}\left(x\right)={2}^{k}{\left(n\right)}_{k}{H}_{n-k}\left(x\right)={2}^{k}\frac{n!}{\left(n-k\right)!}{H}_{n-k}\left(x\right),$
(1.3)

where ${\left(x\right)}_{k}=x\left(x-1\right)\cdots \left(x-k+1\right)$.

As is well known, the Bernoulli polynomials of order r are defined by the generating function to be

${\left(\frac{t}{{e}^{t}-1}\right)}^{r}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n}^{\left(r\right)}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(r\in \mathbb{R}\right).$
(1.4)

In the special case, $x=0$, ${B}_{n}^{\left(r\right)}\left(0\right)={B}_{n}^{\left(r\right)}$ are called the n th Bernoulli numbers of order r (see ).

The Euler polynomials of order r are also defined by the generating function to be

${\left(\frac{2}{{e}^{t}+1}\right)}^{r}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_{n}^{\left(r\right)}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(r\in \mathbb{R}\right).$
(1.5)

In the special case, $x=0$, ${E}_{n}^{\left(r\right)}\left(0\right)={E}_{n}^{\left(r\right)}$ are called the nth Euler numbers of order r.

For $\lambda \left(\ne 1\right)\in \mathbb{C}$, the Frobenius-Euler polynomials of order r are given by

${\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)}^{r}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(r\in \mathbb{R}\right).$
(1.6)

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 nth Frobenius-Euler numbers of order r (see ).

The Stirling numbers of the first kind are defined by the generating function to be

${\left(x\right)}_{n}=\sum _{k=0}^{n}{S}_{1}\left(n,k\right){x}^{k}\phantom{\rule{1em}{0ex}}\text{(see [11, 14])},$
(1.7)

and the Stirling numbers of the second kind are given by

${\left({e}^{t}-1\right)}^{n}=n!\sum _{l=n}^{\mathrm{\infty }}{S}_{2}\left(l,n\right)\frac{{t}^{l}}{l!}\phantom{\rule{1em}{0ex}}\text{(see )}.$
(1.8)

In  it is known that ${H}_{0}\left(x\right),{H}_{1}\left(x\right),\dots ,{H}_{n}\left(x\right)$ from an orthogonal basis for the space

${\mathbb{P}}_{n}=\left\{p\left(x\right)\in \mathbb{Q}\left[x\right]|degp\left(x\right)\le n\right\}$
(1.9)

with respect to the inner product

$〈{p}_{1}\left(x\right),{p}_{2}\left(x\right)〉={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{p}_{1}\left(x\right){p}_{2}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{1em}{0ex}}\text{(see )}.$
(1.10)

For $p\left(x\right)\in {\mathbb{P}}_{n}$, let us assume that

$p\left(x\right)=\sum _{k=0}^{n}{C}_{k}{H}_{k}\left(x\right).$
(1.11)

Then, from the orthogonality of Hermite polynomials and Rodrigues’ formula, we have

$\begin{array}{rcl}{C}_{k}& =& \frac{1}{{2}^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{H}_{k}\left(x\right)p\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& \frac{{\left(-1\right)}^{k}}{{2}^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}}{d{x}^{k}}{e}^{-{x}^{2}}\right)p\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{1em}{0ex}}\text{(see )}.\end{array}$
(1.12)

In particular, for $p\left(x\right)={x}^{m}$ ($m\ge 0$), we easily get

(1.13)

Let be the set of all formal power series in the variable t over with

$\mathcal{F}=\left\{f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{a}_{k}}{k!}{t}^{k}|{a}_{k}\in \mathbb{C}\right\}.$
(1.14)

Let us assume that is the algebra of polynomials in the variable x over and that ${\mathbb{P}}^{\ast }$ is the vector space of all linear functionals on . $〈L|p\left(x\right)〉$ denotes the action of the linear functional L on polynomials $p\left(x\right)$, and we remind that the vector space structure on ${\mathbb{P}}^{\ast }$ is defined by

$\begin{array}{c}〈L+M|p\left(x\right)〉=〈L|p\left(x\right)〉+〈M|p\left(x\right)〉,\hfill \\ 〈cL|p\left(x\right)〉=c〈L|p\left(x\right)〉,\hfill \end{array}$

where c is a complex constant (see [2, 11, 14]).

The formal power series

$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{a}_{k}}{k!}{t}^{k}\in \mathcal{F}$
(1.15)

defines a linear functional on by setting

(1.16)

Thus, by (1.15) and (1.16), we get

$〈{t}^{k}|{x}^{n}〉=n!{\delta }_{n,k}\phantom{\rule{1em}{0ex}}\left(n,k\ge 0\right),$
(1.17)

where ${\delta }_{n,k}$ is the Kronecker symbol (see [2, 11, 14]).

Let ${f}_{L}\left(t\right)={\sum }_{k=0}^{\mathrm{\infty }}\frac{〈L|{x}^{k}〉}{k!}{t}^{k}$. By (1.16), we get

$〈{f}_{L}\left(t\right)|{x}^{n}〉=〈L|{x}^{n}〉,\phantom{\rule{1em}{0ex}}n\ge 0.$
(1.18)

Thus, by (1.18), we see that ${f}_{L}\left(t\right)=L$. The map $L↦{f}_{L}\left(t\right)$ is a vector space isomorphism from ${\mathbb{P}}^{\ast }$ onto . Henceforth, will be thought of as both a formal power series and a linear functional. We call the umbral algebra. The umbral calculus is the study of umbral algebra (see [2, 11, 14]).

The order $o\left(f\left(t\right)\right)$ of the nonzero power series $f\left(t\right)$ is the smallest integer k for which the coefficient of ${t}^{k}$ does not vanish. A series $f\left(t\right)$ having $o\left(f\left(t\right)\right)=1$ is called a delta series, and a series $f\left(t\right)$ having $o\left(f\left(t\right)\right)=0$ is called an invertible series (see [2, 11, 14]). By (1.16) and (1.17), we see that $〈{e}^{yt}|p\left(x\right)〉=p\left(y\right)$. For $f\left(t\right)\in \mathcal{F}$ and $p\left(x\right)\in \mathbb{P}$, we have

$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈f\left(t\right)|{x}^{k}〉}{k!}{t}^{k},\phantom{\rule{2em}{0ex}}p\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈{t}^{k}|p\left(x\right)〉}{k!}{x}^{k}.$
(1.19)

Let $f\left(t\right),g\left(t\right)\in \mathcal{F}$ and $p\left(x\right)\in \mathbb{P}$. Then we easily see that

$〈f\left(t\right)g\left(t\right)|p\left(x\right)〉=〈f\left(t\right)|g\left(t\right)p\left(x\right)〉=〈g\left(t\right)|f\left(t\right)p\left(x\right)〉.$
(1.20)

From (1.19), we can derive the following equation:

${p}^{\left(k\right)}\left(0\right)=〈{t}^{k}|p\left(x\right)〉\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}〈1|{p}^{\left(k\right)}\left(x\right)〉={p}^{\left(k\right)}\left(0\right).$
(1.21)

Thus, by (1.21), we get

${t}^{k}p\left(x\right)={p}^{\left(k\right)}\left(x\right)=\frac{{d}^{k}p\left(x\right)}{d{x}^{k}}\phantom{\rule{1em}{0ex}}\text{(see [2, 11, 14])}.$
(1.22)

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 with $〈g\left(t\right)f{\left(t\right)}^{k}|{S}_{n}\left(x\right)〉=n!{\delta }_{n,k}$, where $n,k\ge 0$ (see [2, 11, 14]). The sequence ${S}_{n}\left(x\right)$ is called Sheffer sequence for $\left(g\left(t\right),f\left(t\right)\right)$, which is denoted by ${S}_{n}\left(x\right)\sim \left(g\left(t\right),f\left(t\right)\right)$. For $f\left(t\right)\in \mathcal{F}$ and $p\left(x\right)\in \mathbb{P}$, we have

$〈\frac{{e}^{yt}-1}{t}|p\left(x\right)〉={\int }_{0}^{y}p\left(u\right)\phantom{\rule{0.2em}{0ex}}du,\phantom{\rule{2em}{0ex}}〈{e}^{yt}-1|p\left(x\right)〉=p\left(y\right)-p\left(0\right),$
(1.23)

and

$〈f\left(t\right)|xp\left(x\right)〉=〈{f}^{\mathrm{\prime }}\left(t\right)|p\left(x\right)〉.$
(1.24)

In this paper, we introduce the identities of several special polynomials which are derived from the orthogonality of Hermite polynomials. Finally, we give some new and interesting identities of the higher-order Bernoulli, Euler and Frobenius-Euler polynomials arising from umbral calculus.

## 2 Some identities of several special polynomials

From (1.5), we note that

${\left(\frac{2}{{e}^{t}+1}\right)}^{r}={\left(1+\frac{{e}^{t}-1}{2}\right)}^{-r}=\sum _{j=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{{e}^{t}-1}{2}\right)}^{j}.$
(2.1)

By (2.1), we get

$\begin{array}{rcl}{\left(\frac{2}{{e}^{t}+1}\right)}^{r}{e}^{xt}& =& \sum _{j=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{{e}^{t}-1}{2}\right)}^{j}{e}^{xt}\\ =& \sum _{n=0}^{\mathrm{\infty }}\left(\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{{e}^{t}-1}{2}\right)}^{j}{x}^{n}\right)\frac{{t}^{n}}{n!}.\end{array}$
(2.2)

From (1.5) and (2.2), we have

${E}_{n}^{\left(r\right)}\left(x\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{-r}{j}\right){2}^{-j}{\left({e}^{t}-1\right)}^{j}{x}^{n}.$
(2.3)

By (1.8) and (1.9), we get

$\begin{array}{rcl}{\left({e}^{t}-1\right)}^{j}{x}^{n}& =& \sum _{k=0}^{n-j}\frac{〈{t}^{k}|{\left({e}^{t}-1\right)}^{j}{x}^{n}〉}{k!}=\sum _{k=0}^{n-j}\frac{〈{\left({e}^{t}-1\right)}^{j}|{t}^{k}{x}^{n}〉}{k!}{x}^{k}\\ =& j!\sum _{k=0}^{n-j}\left(\genfrac{}{}{0}{}{n}{k}\right)\frac{〈{\left({e}^{t}-1\right)}^{j}|{x}^{n-k}〉}{j!}{x}^{k}=j!\sum _{k=0}^{n-j}\left(\genfrac{}{}{0}{}{n}{j}\right){S}_{2}\left(n-k,j\right){x}^{k}\\ =& j!\sum _{k=j}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){S}_{2}\left(k,j\right){x}^{n-k}.\end{array}$
(2.4)

Therefore, by (2.3) and (2.4), we obtain the following theorem.

Theorem 2.1 For $n\ge 0$, we have

$\begin{array}{rcl}{E}_{n}^{\left(r\right)}\left(x\right)& =& \sum _{0\le j\le n}\sum _{j\le k\le n}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{2}^{j}}{S}_{2}\left(k,j\right){x}^{n-k}\\ =& \sum _{0\le k\le n}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[\sum _{0\le j\le k}\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{2}^{j}}{S}_{2}\left(k,j\right)\right]{x}^{n-k}.\end{array}$

By (1.5), we easily see that

${E}_{n}^{\left(r\right)}\left(x\right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){E}_{k}^{\left(r\right)}{x}^{n-k}.$
(2.5)

Therefore, by Theorem 2.1 and (2.5), we obtain the following corollary.

Corollary 2.2 For $k\ge 0$, we have

${E}_{k}^{\left(r\right)}=\sum _{j=0}^{k}\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{2}^{j}}{S}_{2}\left(k,j\right).$

Let us take $p\left(x\right)={E}_{n}^{\left(r\right)}\left(x\right)\in {\mathbb{P}}_{n}$. Then, by (1.11), we get

${E}_{n}^{\left(r\right)}\left(x\right)=\sum _{k=0}^{n}{C}_{k}{H}_{k}\left(x\right).$
(2.6)

From (1.12), we can derive the computation of ${C}_{k}$ as follows:

${C}_{k}=\frac{{\left(-1\right)}^{k}}{{2}^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){E}_{n}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,$
(2.7)

where

$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){E}_{n}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}=\left(-n\right)\left(-\left(n-1\right)\right)\cdots \left(-\left(n-k+1\right)\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{E}_{n-k}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(-1\right)}^{k}n!}{\left(n-k\right)!}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0}{}{n-k}{l}\right){E}_{n-k-l}^{\left(r\right)}{x}^{l}\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(-1\right)}^{k}n!}{\left(n-k\right)!}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0}{}{n-k}{l}\right){E}_{n-k-l}^{\left(r\right)}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{x}^{l}\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}={\left(-1\right)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{1}{\left(n-k-l\right)!{2}^{l}\left(\frac{l}{2}\right)!}\sum _{j=0}^{n-k-l}\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{2}^{j}}{S}_{2}\left(n-k-l,j\right).\hfill \end{array}$
(2.8)

From (2.7) and (2.8), we can derive the following equation:

$\begin{array}{rcl}{C}_{k}& =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{E}_{n-k-l}^{\left(r\right)}}{k!\left(n-k-l\right)!{2}^{k+l}\left(\frac{l}{2}\right)!}\\ =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0}{}{-r}{j}\right)j!{S}_{2}\left(n-k-l,j\right)}{k!\left(n-k-l\right)!{2}^{k+l+j}\left(\frac{l}{2}\right)!}.\end{array}$
(2.9)

Therefore, by Corollary 2.2, (2.6) and (2.9), we obtain the following theorem.

Theorem 2.3 For $n\ge 0$, we have

$\begin{array}{rcl}{E}_{n}^{\left(r\right)}\left(x\right)& =& n!\sum _{k=0}^{n}\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{E}_{n-k-l}^{\left(r\right)}}{k!\left(n-k-l\right)!{2}^{k+l}\left(\frac{l}{2}\right)!}\right\}{H}_{k}\left(x\right)\\ =& n!\sum _{k=0}^{n}\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0}{}{-r}{j}\right)j!{S}_{2}\left(n-k-l,j\right)}{k!\left(n-k-l\right)!{2}^{k+l+j}\left(\frac{l}{2}\right)!}\right\}{H}_{k}\left(x\right).\end{array}$

By (1.4), we easily see that

${\left(\frac{t}{{e}^{t}-1}\right)}^{r}={\left(1+\frac{{e}^{t}-t-1}{t}\right)}^{-r}=\sum _{j=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}.$
(2.10)

Thus, by (2.10), we get

${\left(\frac{t}{{e}^{t}-1}\right)}^{r}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}{x}^{n}\right)\frac{{t}^{n}}{n!}.$
(2.11)

From (1.4) and (2.11), we have

${B}_{n}^{\left(r\right)}\left(x\right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}{x}^{n}.$
(2.12)

By (1.19), we easily get

$\begin{array}{rcl}{\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}{x}^{n}& =& \sum _{k=0}^{n-j}\frac{〈{t}^{k}|{\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}{x}^{n}〉}{k!}{x}^{k}\\ =& \sum _{k=0}^{n-j}\frac{〈{\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}|{t}^{k}{x}^{n}〉}{k!}{x}^{k}=\sum _{k=0}^{n-j}\left(\genfrac{}{}{0}{}{n}{k}\right)\sum _{l=0}^{j}\left(\genfrac{}{}{0}{}{j}{l}\right){\left(-1\right)}^{j-l}〈{\left(\frac{{e}^{t}-1}{t}\right)}^{l}|{x}^{n-k}〉{x}^{k}\\ =& \sum _{k=0}^{n-j}\left(\genfrac{}{}{0}{}{n}{k}\right)\sum _{l=0}^{j}\left(\genfrac{}{}{0}{}{j}{l}\right){\left(-1\right)}^{j-l}〈{t}^{0}|{\left(\frac{{e}^{t}-1}{t}\right)}^{l}{x}^{n-k}〉{x}^{k}.\end{array}$
(2.13)

From (1.8), (1.21) and (2.13), we have

${\left(\frac{{e}^{t}-t-1}{t}\right)}^{j}{x}^{n}=\sum _{k=0}^{n-j}\sum _{l=0}^{j}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{j}{l}\right){\left(-1\right)}^{j-l}\frac{\left(n-k\right)!l!}{\left(n-k+l\right)!}{S}_{2}\left(n-k+l,l\right){x}^{k}.$
(2.14)

Thus, by (2.12) and (2.14), we get

$\begin{array}{rcl}{B}_{n}^{\left(r\right)}\left(x\right)& =& \sum _{j=0}^{n}\sum _{k=0}^{n-j}\sum _{l=0}^{j}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{j}{l}\right){\left(-1\right)}^{j-l}\frac{{S}_{2}\left(n-k+l,l\right)}{\left(\genfrac{}{}{0}{}{n-k+l}{l}\right)}{x}^{k}\\ =& \sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[\sum _{j=0}^{k}\sum _{l=0}^{j}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{j}{l}\right)\frac{{S}_{2}\left(k+l,l\right)}{\left(\genfrac{}{}{0}{}{k+l}{l}\right)}{\left(-1\right)}^{j-l}\right]{x}^{n-k}.\end{array}$
(2.15)

Therefore, by (2.12) and (2.15), we obtain the following theorem.

Theorem 2.4 For $n\ge 0$, we have

${B}_{n}^{\left(r\right)}\left(x\right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[\sum _{j=0}^{k}\sum _{l=0}^{j}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{j}{l}\right)\frac{{S}_{2}\left(k+l,l\right)}{\left(\genfrac{}{}{0}{}{k+l}{l}\right)}{\left(-1\right)}^{j-l}\right]{x}^{n-k}.$

By (1.4), we easily get

${B}_{n}^{\left(r\right)}\left(x\right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){B}_{k}^{\left(r\right)}{x}^{n-k}.$
(2.16)

Therefore, by Theorem 2.4 and (2.16), we obtain the following corollary.

Corollary 2.5 For $k\ge 0$, we have

${B}_{k}^{\left(r\right)}=\sum _{j=0}^{k}\sum _{l=0}^{j}{\left(-1\right)}^{j-l}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{j}{l}\right)\frac{{S}_{2}\left(k+l,l\right)}{\left(\genfrac{}{}{0}{}{k+l}{l}\right)}.$

Let us consider $p\left(x\right)={B}_{n}^{\left(r\right)}\left(x\right)\in {\mathbb{P}}_{n}$. Then, by (1.11), ${B}_{n}^{\left(r\right)}\left(x\right)$ can be written as

${B}_{n}^{\left(r\right)}\left(x\right)=\sum _{k=0}^{n}{C}_{k}{H}_{k}\left(x\right).$
(2.17)

Now, we compute ${C}_{k}$’s for ${B}_{k}^{\left(r\right)}\left(x\right)$ as follows:

${C}_{k}=\frac{{\left(-1\right)}^{k}}{{2}^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){B}_{n}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,$
(2.18)

where

$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){B}_{n}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}=\left(-n\right)\left(-\left(n-1\right)\right)\cdots \left(-\left(n-k+1\right)\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{B}_{n-k}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(-1\right)}^{k}n!}{\left(n-k\right)!}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0}{}{n-k}{l}\right){B}_{n-k-l}^{\left(r\right)}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{x}^{l}\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}={\left(-1\right)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{B}_{n-k-l}^{\left(r\right)}}{\left(n-k-l\right)!{2}^{l}\left(\frac{l}{2}\right)!}.\hfill \end{array}$
(2.19)

By Corollary 2.5 and (2.19), we get

$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){B}_{n}^{\left(r\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}={\left(-1\right)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}s\sum _{j=0}^{n-k-l}\sum _{m=0}^{j}\frac{{\left(-1\right)}^{j-m}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{j}{m}\right){S}_{2}\left(n-k-l+m,m\right)}{\left(n-k-l\right)!{2}^{l}\left(\frac{l}{2}\right)!\left(\genfrac{}{}{0}{}{n-k-l+m}{m}\right)}.\hfill \end{array}$
(2.20)

From (2.18) and (2.20), we have

$\begin{array}{rcl}{C}_{k}& =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{B}_{n-k-l}^{\left(r\right)}}{\left(n-k-l\right)!k!{2}^{k+l}\left(\frac{l}{2}\right)!}\\ =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\sum _{m=0}^{j}\frac{{\left(-1\right)}^{j-m}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{j}{m}\right){S}_{2}\left(n-k-l+m,m\right)}{\left(n-k-l\right)!k!{2}^{k+l}\left(\frac{l}{2}\right)!\left(\genfrac{}{}{0}{}{n-k-l+m}{m}\right)}.\end{array}$
(2.21)

Therefore, by (2.17) and (2.21), we obtain the following theorem.

Theorem 2.6 For $n\ge 0$, we have

$\begin{array}{rcl}{B}_{n}^{\left(r\right)}\left(x\right)& =& n!\sum _{k=0}^{n}\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{B}_{n-k-l}^{\left(r\right)}}{\left(n-k-l\right)!k!{2}^{k+l}\left(\frac{l}{2}\right)!}\right\}{H}_{k}\left(x\right)\\ =& n!\sum _{k=0}^{n}\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\sum _{m=0}^{j}\frac{{\left(-1\right)}^{j-m}\left(\genfrac{}{}{0}{}{-r}{j}\right)\left(\genfrac{}{}{0}{}{j}{m}\right){S}_{2}\left(n-k-l+m,m\right)}{\left(n-k-l\right)!k!{2}^{k+l}\left(\frac{l}{2}\right)!\left(\genfrac{}{}{0}{}{n-k-l+m}{m}\right)}\right\}{H}_{k}\left(x\right).\end{array}$

It is easy to show that

${\left(\frac{1-\lambda }{{e}^{t}-\lambda }\right)}^{r}={\left(1+\frac{{e}^{t}-1}{1-\lambda }\right)}^{-r}=\sum _{j=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(\frac{1}{1-\lambda }\right)}^{j}{\left({e}^{t}-1\right)}^{j}.$
(2.22)

From (1.6) and (2.22), we have

${H}_{n}^{\left(r\right)}\left(x|\lambda \right)=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{-r}{j}\right){\left(1-\lambda \right)}^{-j}{\left({e}^{t}-1\right)}^{j}{x}^{n},$
(2.23)

where

$\begin{array}{rcl}{\left({e}^{t}-1\right)}^{j}{x}^{n}& =& j!\sum _{k=j}^{\mathrm{\infty }}{S}_{2}\left(k,j\right)\frac{{t}^{k}}{k!}{x}^{n}\\ =& j!\sum _{k=j}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){S}_{2}\left(k,j\right){x}^{n-k}.\end{array}$
(2.24)

Thus, by (2.24), we get

${\left({e}^{t}-1\right)}^{j}{x}^{n}=j!\sum _{k=j}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){S}_{2}\left(k,j\right){x}^{n-k}.$
(2.25)

From (2.23) and (2.25), we can derive the following equation:

$\begin{array}{rcl}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)& =& \sum _{j=0}^{n}\sum _{k=j}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{\left(1-\lambda \right)}^{j}}{S}_{2}\left(k,j\right){x}^{n-k}\\ =& \sum _{k=0}^{n}\sum _{j=0}^{k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{\left(1-\lambda \right)}^{j}}{S}_{2}\left(k,j\right){x}^{n-k}\\ =& \sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[\sum _{j=0}^{k}\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{\left(1-\lambda \right)}^{j}}{S}_{2}\left(k,j\right)\right]{x}^{n-k}.\end{array}$
(2.26)

By (1.6), we easily see that

${H}_{n}^{\left(r\right)}\left(x|\lambda \right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){H}_{k}^{\left(r\right)}\left(\lambda \right){x}^{n-k}.$
(2.27)

Therefore, by (2.26) and (2.27), we obtain the following theorem.

Theorem 2.7 For $k\ge 0$, we have

${H}_{k}^{\left(r\right)}\left(\lambda \right)=\sum _{j=0}^{k}\left(\genfrac{}{}{0}{}{-r}{j}\right)\frac{j!}{{\left(1-\lambda \right)}^{j}}{S}_{2}\left(k,j\right).$

Let us take $p\left(x\right)={H}_{n}^{\left(r\right)}\left(x|\lambda \right)\in {\mathbb{P}}_{n}$. Then, by (1.11), ${H}_{n}^{\left(r\right)}\left(x|\lambda \right)$ is given by

${H}_{n}^{\left(r\right)}\left(x|\lambda \right)=\sum _{k=0}^{n}{C}_{k}{H}_{k}\left(x\right).$
(2.28)

By (1.12), we get

${C}_{k}=\frac{{\left(-1\right)}^{k}}{{2}^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){H}_{n}^{\left(r\right)}\left(x|\lambda \right)\phantom{\rule{0.2em}{0ex}}dx,$
(2.29)

where

$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{d}^{k}{e}^{-{x}^{2}}}{d{x}^{k}}\right){H}_{n}^{\left(r\right)}\left(x|\lambda \right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(-1\right)}^{k}n!}{\left(n-k\right)!}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0}{}{n-k}{l}\right){H}_{n-k-l}^{\left(r\right)}\left(\lambda \right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-{x}^{2}}{x}^{l}\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}={\left(-1\right)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{H}_{n-k-l}^{\left(r\right)}\left(\lambda \right)}{\left(n-k-l\right)!{2}^{l}\left(\frac{l}{2}\right)!}\hfill \\ \phantom{\rule{1em}{0ex}}={\left(-1\right)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0}{}{-r}{j}\right)j!{S}_{2}\left(n-k-l,j\right)}{\left(n-k-l\right)!{2}^{l}{\left(1-\lambda \right)}^{j}\left(\frac{l}{2}\right)!}.\hfill \end{array}$
(2.30)

By (2.29) and (2.30), we get

$\begin{array}{rcl}{C}_{k}& =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{H}_{n-k-l}^{\left(r\right)}\left(\lambda \right)}{\left(n-k-l\right)!k!{2}^{l+k}\left(\frac{l}{2}\right)!}\\ =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0}{}{-r}{j}\right)j!{S}_{2}\left(n-k-l,j\right)}{\left(n-k-l\right)!k!{2}^{k+l}{\left(1-\lambda \right)}^{j}\left(\frac{l}{2}\right)!}.\end{array}$
(2.31)

Therefore, by (2.28) and (2.31), we obtain the following theorem.

Corollary 2.8 For $n\ge 0$, we have

$\begin{array}{rcl}{H}_{n}^{\left(r\right)}\left(x|\lambda \right)& =& n!\sum _{k=0}^{n}\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{{H}_{n-k-l}^{\left(r\right)}\left(\lambda \right)}{\left(n-k-l\right)!k!{2}^{l+k}\left(\frac{l}{2}\right)!}\right\}{H}_{k}\left(x\right)\\ =& n!\sum _{k=0}^{n}\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0}{}{-r}{j}\right)j!{S}_{2}\left(n-k-l,j\right)}{\left(n-k-l\right)!k!{2}^{k+l}{\left(1-\lambda \right)}^{j}\left(\frac{l}{2}\right)!}\right\}{H}_{k}\left(x\right).\end{array}$

## References

1. Kim DS, Kim T, Rim SH, Lee S-H: Hermite polynomials and their applications associated with Bernoulli and Euler numbers. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 974632, 13 pp. doi:10.1155/2012/974632

2. Kim T: Symmetry p -adic invariant integral on ${\mathbb{Z}}_{p}$ for Bernoulli and Euler polynomials. J. Differ. Equ. Appl. 2008, 14: 1267–1277. 10.1080/10236190801943220

3. Kim T, Choi J, Kim YH, Ryoo CS: On q -Bernstein and q -Hermite polynomials. Proc. Jangjeon Math. Soc. 2011, 14(2):215–221.

4. Ryoo C: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):217–223.

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

6. Araci S, Erdal D, Seo J: A study on the fermionic p -adic q -integral representation on ${\mathbb{Z}}_{p}$ associated with weighted q -Bernstein and q -Genocchi polynomials. Abstr. Appl. Anal. 2011., 2011: Article ID 649248, 10 pp.

7. Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.

8. Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler-functions. Adv. Stud. Contemp. Math. 2009, 18(2):135–160.

9. Carlitz L: The product of two Eulerian polynomials. Math. Mag. 1963, 368: 37–41.

10. Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.

11. Kim DS, Kim T: Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv. Differ. Equ. 2012., 2012: Article ID 196. doi:10.1186/1687–1847–2012–196

12. Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41–48.

13. Rim S-H, Joung J, Jin J-H, Lee S-J: A note on the weighted Carlitz’s type q -Euler numbers and q -Bernstein polynomials. Proc. Jangjeon Math. Soc. 2012, 15: 195–201.

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

15. Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on ${\mathbb{Z}}_{p}$ . Rocky Mt. J. Math. 2011, 41: 239–247. 10.1216/RMJ-2011-41-1-239

16. Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251–278.

## Acknowledgements

The authors would like to express their sincere gratitude to referees for their valuable comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2013.

## Author information

Authors

### Corresponding author

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.

## Rights and permissions

Reprints and Permissions

Kim, D.S., Kim, T., Dolgy, D.V. et al. Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus. Adv Differ Equ 2013, 73 (2013). https://doi.org/10.1186/1687-1847-2013-73 