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

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

(1.1)

with the usual convention about replacing $H^n(x)$ by $H_n(x)$ (see [1]). In the special case, $x=0$, $H_n(0)=H_n$ are called the nth Hermite numbers. From (1.1) we have

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

(1.2)

Thus, by (1.2), we get

$$\frac{d^k}{d{x}^k}{H}_n(x)=2^k{(n)}_k{H}_{n-k}(x)=2^k\frac{n!}{(n-k)!}{H}_{n-k}(x),$$

(1.3)

where ${(x)}_k=x(x-1)\cdots (x-k+1)$.

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

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

(1.4)

In the special case, $x=0$, $B_n^{(r)}(0)=B_n^{(r)}$ are called the n th Bernoulli numbers of order r (see [1–4]).

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

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

(1.5)

In the special case, $x=0$, $E_n^{(r)}(0)=E_n^{(r)}$ are called the nth Euler numbers of order r.

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

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

(1.6)

In the special case, $x=0$, $H_n^{(r)}(0|\lambda )=H_n^{(r)}(\lambda )$ are called the nth Frobenius-Euler numbers of order r (see [1–16]).

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

$${(x)}_n=\sum _{k=0}^n{S}_1(n,k)x^k\text{(see [11, 14])},$$

(1.7)

and the Stirling numbers of the second kind are given by

$${(e^t-1)}^n=n!\sum _{l=n}^{\mathrm{\infty }}{S}_2(l,n)\frac{t^l}{l!}\text{(see [14])}.$$

(1.8)

In [1] it is known that $H_0(x),H_1(x),\dots ,H_n(x)$ from an orthogonal basis for the space

$${\mathbb{P}}_n=\{p(x)\in \mathbb{Q}[x]|degp(x)\le n\}$$

(1.9)

with respect to the inner product

$$〈p_1(x),p_2(x)〉={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}{p}_1(x)p_2(x)dx\text{(see [1])}.$$

(1.10)

For $p(x)\in {\mathbb{P}}_n$, let us assume that

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

(1.11)

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

$$\begin{array}{rcl}{C}_k& =& \frac{1}{2^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}{H}_k(x)p(x)dx\\ =& \frac{{(-1)}^k}{2^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k}{d{x}^k}{e}^{-x^2}\right)p(x)dx\text{(see [1])}.\end{array}$$

(1.12)

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

$$\begin{array}{r}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^n}{d{x}^n}{e}^{-x^2}\right)x^{m}dx\\ =\{\begin{array}{cc}0\hfill & \text{if }n>m\text{ or }n\le m\text{ with }m-n\not\equiv 0(mod2),\hfill \\ \frac{{(-1)}^{n}m!\sqrt{\pi }}{2^{m-n}(\frac{m-n}{2})!}\hfill & \text{if }n\le m\text{ with }m-n\equiv 0(mod2).\hfill \end{array}\end{array}$$

(1.13)

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

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

(1.14)

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

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

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

The formal power series

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

(1.15)

defines a linear functional on ℙ by setting

$$〈f(t)|x^n〉=a_n\text{for all }n\in {\mathbb{Z}}_{+}=\mathbb{N}\cup \{0\}.$$

(1.16)

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

$$〈t^k|x^n〉=n!{\delta }_{n,k}(n,k\ge 0),$$

(1.17)

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

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

$$〈f_L(t)|x^n〉=〈L|x^n〉,n\ge 0.$$

(1.18)

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

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

$$f(t)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈f(t)|x^k〉}{k!}{t}^k,p(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈t^k|p(x)〉}{k!}{x}^k.$$

(1.19)

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

$$〈f(t)g(t)|p(x)〉=〈f(t)|g(t)p(x)〉=〈g(t)|f(t)p(x)〉.$$

(1.20)

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

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

(1.21)

Thus, by (1.21), we get

$$t^{k}p(x)=p^{(k)}(x)=\frac{d^{k}p(x)}{d{x}^k}\text{(see [2, 11, 14])}.$$

(1.22)

Let $f(t)$ be a delta series, and let $g(t)$ be an invertible series. Then there exists a unique sequence $S_n(x)$ of polynomials with $〈g(t)f{(t)}^k|S_n(x)〉=n!{\delta }_{n,k}$, where $n,k\ge 0$ (see [2, 11, 14]). The sequence $S_n(x)$ is called Sheffer sequence for $(g(t),f(t))$, which is denoted by $S_n(x)\sim (g(t),f(t))$. For $f(t)\in \mathcal{F}$ and $p(x)\in \mathbb{P}$, we have

$$〈\frac{e^{yt}-1}{t}|p(x)〉={\int }_0^{y}p(u)du,〈e^{yt}-1|p(x)〉=p(y)-p(0),$$

(1.23)

and

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

(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

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

(2.1)

By (2.1), we get

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

(2.2)

From (1.5) and (2.2), we have

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

(2.3)

By (1.8) and (1.9), we get

$$\begin{array}{rcl}{(e^t-1)}^j{x}^n& =& \sum _{k=0}^{n-j}\frac{〈t^k|{(e^t-1)}^j{x}^n〉}{k!}=\sum _{k=0}^{n-j}\frac{〈{(e^t-1)}^j|t^k{x}^n〉}{k!}{x}^k\\ =& j!\sum _{k=0}^{n-j}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{〈{(e^t-1)}^j|x^{n-k}〉}{j!}{x}^k=j!\sum _{k=0}^{n-j}\left(\genfrac{}{}{0ex}{}{n}{j}\right)S_2(n-k,j)x^k\\ =& j!\sum _{k=j}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)S_2(k,j)x^{n-k}.\end{array}$$

(2.4)

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

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

$$\begin{array}{rcl}{E}_n^{(r)}(x)& =& \sum _{0\le j\le n}\sum _{j\le k\le n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{2^j}{S}_2(k,j)x^{n-k}\\ =& \sum _{0\le k\le n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)[\sum _{0\le j\le k}\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{2^j}{S}_2(k,j)]x^{n-k}.\end{array}$$

By (1.5), we easily see that

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

(2.5)

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

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

$$E_k^{(r)}=\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{2^j}{S}_2(k,j).$$

Let us take $p(x)=E_n^{(r)}(x)\in {\mathbb{P}}_n$. Then, by (1.11), we get

$$E_n^{(r)}(x)=\sum _{k=0}^n{C}_k{H}_k(x).$$

(2.6)

From (1.12), we can derive the computation of $C_k$ as follows:

$$C_k=\frac{{(-1)}^k}{2^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k{e}^{-x^2}}{d{x}^k}\right)E_n^{(r)}(x)dx,$$

(2.7)

where

$$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k{e}^{-x^2}}{d{x}^k}\right)E_n^{(r)}(x)dx\hfill \\ =(-n)(-(n-1))\cdots (-(n-k+1)){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}{E}_{n-k}^{(r)}(x)dx\hfill \\ =\frac{{(-1)}^{k}n!}{(n-k)!}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0ex}{}{n-k}{l}\right)E_{n-k-l}^{(r)}{x}^{l}dx\hfill \\ =\frac{{(-1)}^{k}n!}{(n-k)!}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0ex}{}{n-k}{l}\right)E_{n-k-l}^{(r)}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}{x}^{l}dx\hfill \\ ={(-1)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{1}{(n-k-l)!2^l(\frac{l}{2})!}\sum _{j=0}^{n-k-l}\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{2^j}{S}_2(n-k-l,j).\hfill \end{array}$$

(2.8)

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

$$\begin{array}{rcl}{C}_k& =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{E_{n-k-l}^{(r)}}{k!(n-k-l)!2^{k+l}(\frac{l}{2})!}\\ =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0ex}{}{-r}{j}\right)j!S_2(n-k-l,j)}{k!(n-k-l)!2^{k+l+j}(\frac{l}{2})!}.\end{array}$$

(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

$$\begin{array}{rcl}{E}_n^{(r)}(x)& =& n!\sum _{k=0}^n\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{E_{n-k-l}^{(r)}}{k!(n-k-l)!2^{k+l}(\frac{l}{2})!}\right\}{H}_k(x)\\ =& n!\sum _{k=0}^n\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0ex}{}{-r}{j}\right)j!S_2(n-k-l,j)}{k!(n-k-l)!2^{k+l+j}(\frac{l}{2})!}\right\}{H}_k(x).\end{array}$$

By (1.4), we easily see that

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

(2.10)

Thus, by (2.10), we get

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

(2.11)

From (1.4) and (2.11), we have

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

(2.12)

By (1.19), we easily get

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

(2.13)

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

$${\left(\frac{e^t-t-1}{t}\right)}^j{x}^n=\sum _{k=0}^{n-j}\sum _{l=0}^j\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{j}{l}\right){(-1)}^{j-l}\frac{(n-k)!l!}{(n-k+l)!}{S}_2(n-k+l,l)x^k.$$

(2.14)

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

$$\begin{array}{rcl}{B}_n^{(r)}(x)& =& \sum _{j=0}^n\sum _{k=0}^{n-j}\sum _{l=0}^j\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{j}{l}\right){(-1)}^{j-l}\frac{S_2(n-k+l,l)}{\left(\genfrac{}{}{0ex}{}{n-k+l}{l}\right)}{x}^k\\ =& \sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left[\sum _{j=0}^k\sum _{l=0}^j\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{l}\right)\frac{S_2(k+l,l)}{\left(\genfrac{}{}{0ex}{}{k+l}{l}\right)}{(-1)}^{j-l}\right]x^{n-k}.\end{array}$$

(2.15)

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

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

$$B_n^{(r)}(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left[\sum _{j=0}^k\sum _{l=0}^j\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{l}\right)\frac{S_2(k+l,l)}{\left(\genfrac{}{}{0ex}{}{k+l}{l}\right)}{(-1)}^{j-l}\right]x^{n-k}.$$

By (1.4), we easily get

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

(2.16)

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

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

$$B_k^{(r)}=\sum _{j=0}^k\sum _{l=0}^j{(-1)}^{j-l}\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{l}\right)\frac{S_2(k+l,l)}{\left(\genfrac{}{}{0ex}{}{k+l}{l}\right)}.$$

Let us consider $p(x)=B_n^{(r)}(x)\in {\mathbb{P}}_n$. Then, by (1.11), $B_n^{(r)}(x)$ can be written as

$$B_n^{(r)}(x)=\sum _{k=0}^n{C}_k{H}_k(x).$$

(2.17)

Now, we compute $C_k$’s for $B_k^{(r)}(x)$ as follows:

$$C_k=\frac{{(-1)}^k}{2^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k{e}^{-x^2}}{d{x}^k}\right)B_n^{(r)}(x)dx,$$

(2.18)

where

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

(2.19)

By Corollary 2.5 and (2.19), we get

$$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k{e}^{-x^2}}{d{x}^k}\right)B_n^{(r)}(x)dx\hfill \\ ={(-1)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}s\sum _{j=0}^{n-k-l}\sum _{m=0}^j\frac{{(-1)}^{j-m}\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{m}\right)S_2(n-k-l+m,m)}{(n-k-l)!2^l(\frac{l}{2})!\left(\genfrac{}{}{0ex}{}{n-k-l+m}{m}\right)}.\hfill \end{array}$$

(2.20)

From (2.18) and (2.20), we have

$$\begin{array}{rcl}{C}_k& =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{B_{n-k-l}^{(r)}}{(n-k-l)!k!2^{k+l}(\frac{l}{2})!}\\ =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\sum _{m=0}^j\frac{{(-1)}^{j-m}\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{m}\right)S_2(n-k-l+m,m)}{(n-k-l)!k!2^{k+l}(\frac{l}{2})!\left(\genfrac{}{}{0ex}{}{n-k-l+m}{m}\right)}.\end{array}$$

(2.21)

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

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

$$\begin{array}{rcl}{B}_n^{(r)}(x)& =& n!\sum _{k=0}^n\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{B_{n-k-l}^{(r)}}{(n-k-l)!k!2^{k+l}(\frac{l}{2})!}\right\}{H}_k(x)\\ =& n!\sum _{k=0}^n\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\sum _{m=0}^j\frac{{(-1)}^{j-m}\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{m}\right)S_2(n-k-l+m,m)}{(n-k-l)!k!2^{k+l}(\frac{l}{2})!\left(\genfrac{}{}{0ex}{}{n-k-l+m}{m}\right)}\right\}{H}_k(x).\end{array}$$

It is easy to show that

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

(2.22)

From (1.6) and (2.22), we have

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

(2.23)

where

$$\begin{array}{rcl}{(e^t-1)}^j{x}^n& =& j!\sum _{k=j}^{\mathrm{\infty }}{S}_2(k,j)\frac{t^k}{k!}{x}^n\\ =& j!\sum _{k=j}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)S_2(k,j)x^{n-k}.\end{array}$$

(2.24)

Thus, by (2.24), we get

$${(e^t-1)}^j{x}^n=j!\sum _{k=j}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)S_2(k,j)x^{n-k}.$$

(2.25)

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

$$\begin{array}{rcl}{H}_n^{(r)}(x|\lambda )& =& \sum _{j=0}^n\sum _{k=j}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{{(1-\lambda )}^j}{S}_2(k,j)x^{n-k}\\ =& \sum _{k=0}^n\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{{(1-\lambda )}^j}{S}_2(k,j)x^{n-k}\\ =& \sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)[\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{{(1-\lambda )}^j}{S}_2(k,j)]x^{n-k}.\end{array}$$

(2.26)

By (1.6), we easily see that

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

(2.27)

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

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

$$H_k^{(r)}(\lambda )=\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{-r}{j}\right)\frac{j!}{{(1-\lambda )}^j}{S}_2(k,j).$$

Let us take $p(x)=H_n^{(r)}(x|\lambda )\in {\mathbb{P}}_n$. Then, by (1.11), $H_n^{(r)}(x|\lambda )$ is given by

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

(2.28)

By (1.12), we get

$$C_k=\frac{{(-1)}^k}{2^{k}k!\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k{e}^{-x^2}}{d{x}^k}\right)H_n^{(r)}(x|\lambda )dx,$$

(2.29)

where

$$\begin{array}{c}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{d^k{e}^{-x^2}}{d{x}^k}\right)H_n^{(r)}(x|\lambda )dx\hfill \\ =\frac{{(-1)}^{k}n!}{(n-k)!}\sum _{l=0}^{n-k}\left(\genfrac{}{}{0ex}{}{n-k}{l}\right)H_{n-k-l}^{(r)}(\lambda ){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}{x}^{l}dx\hfill \\ ={(-1)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{H_{n-k-l}^{(r)}(\lambda )}{(n-k-l)!2^l(\frac{l}{2})!}\hfill \\ ={(-1)}^{k}n!\sqrt{\pi }\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0ex}{}{-r}{j}\right)j!S_2(n-k-l,j)}{(n-k-l)!2^l{(1-\lambda )}^j(\frac{l}{2})!}.\hfill \end{array}$$

(2.30)

By (2.29) and (2.30), we get

$$\begin{array}{rcl}{C}_k& =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{H_{n-k-l}^{(r)}(\lambda )}{(n-k-l)!k!2^{l+k}(\frac{l}{2})!}\\ =& n!\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0ex}{}{-r}{j}\right)j!S_2(n-k-l,j)}{(n-k-l)!k!2^{k+l}{(1-\lambda )}^j(\frac{l}{2})!}.\end{array}$$

(2.31)

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

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

$$\begin{array}{rcl}{H}_n^{(r)}(x|\lambda )& =& n!\sum _{k=0}^n\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\frac{H_{n-k-l}^{(r)}(\lambda )}{(n-k-l)!k!2^{l+k}(\frac{l}{2})!}\right\}{H}_k(x)\\ =& n!\sum _{k=0}^n\left\{\sum _{0\le l\le n-k,l:\mathrm{even}}\sum _{j=0}^{n-k-l}\frac{\left(\genfrac{}{}{0ex}{}{-r}{j}\right)j!S_2(n-k-l,j)}{(n-k-l)!k!2^{k+l}{(1-\lambda )}^j(\frac{l}{2})!}\right\}{H}_k(x).\end{array}$$