Formulae for Generalization of Touchard Polynomials with Their Generating Functions

Abstract

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given.

1. Introduction

Special polynomial and numbers have been used many useful applications in many applied areas. Therefore, the motivation of this paper is to give many novel formulas, relations for the generalization of the Touchard polynomials. It is known that the Touchard polynomials, defined by Touchard in 1939, are a member of the family of combinatorial polynomials. These polynomials are also known as the Bell polynomials or exponential polynomials (cf. [1,2,3,4,5,6,7,8]).

Berndt ([2], Entry 4, p. 47) gave the following generating function for the Touchard polynomials $T_m(u);n\in \mathbf{N}=\{1,2,\dots \}$ and $u\in \mathbf{R}$:

$$F_T(t,u)=e^{u\left(e^t-1\right)}=\sum _{m=0}^{\infty }{T}_m(u){\displaystyle \frac{t^m}{m!}},$$

(1)

(cf. [1], see also [5,8,9,10]). By using (1), we have

$$\sum _{n=0}^{\infty }{u}^n{F}_S(t,n)=\sum _{m=0}^{\infty }{T}_m(u){\displaystyle \frac{t^m}{m!}}.$$

where

$$F_S(t,n)={\displaystyle \frac{{(e^t-1)}^n}{n!}}=\sum _{m=0}^{\infty }{S}_2(m,n){\displaystyle \frac{t^m}{m!}}$$

where $S_2(n,j)$ denotes the Stirling numbers of the second kind (cf. [8]). Thus, we get

$$\sum _{m=0}^{\infty }\sum _{n=0}^m{u}^n{S}_2(m,n){\displaystyle \frac{t^m}{m!}}=\sum _{m=0}^{\infty }{T}_m(u){\displaystyle \frac{t^m}{m!}}.$$

Comparing coefficient ${\displaystyle \frac{t^m}{m!}}$ on the both sides of the above equation, we have for the polynomials $T_m(u)$ (cf. [2]):

$$T_m(u)=\sum _{n=0}^m{u}^n{S}_2(m,n)$$

(2)

where $m\in \mathbf{N}$ and $T_m(1)=B{l}_m$, denoted Bell numbers, which are given by

$$B{l}_m=\sum _{n=0}^m{S}_2(m,n)$$

(cf. [1,2]).

Ramanujan also showed that Touchard polynomials $T_m(u)$ appear in the asymptotic expansion of a meromorphic function $\mathsf{\Psi }(t;u)$, details of which are given as follows:

$$\mathsf{\Psi }(t;u)=\sum _{c=1}^{\infty }{\displaystyle \frac{{(-1)}^{c-1}{u}^c}{{\displaystyle \prod _{j=1}^c}\left(t+j\right)}}$$

where u is any complex number with $u\ne 0$:

$$\mathsf{\Psi }(t;u)\sim \sum _{c=1}^{\infty }{\displaystyle \frac{{(-1)}^{c-1}}{t^m}}{T}_{m-1}(u).$$

For more detailed information on this subject, refer to Berndt’s book ([2], p. 47).

Inspired by Ramanujan’s work, Berndt ([2], Eq-(6.1)) gave the following interesting formula for the Touchard polynomials:

$$T_m(u)=\sum _{j=1}^m\mathsf{\Omega }(m)u^j$$

where the integers $\mathsf{\Omega }(m)$ are defined by

$$\sum _{c=1}^{\infty }{\displaystyle \frac{c^m{u}^c}{(c-1)!}}=e^u\sum _{c=1}^{m+1}\mathsf{\Omega }(c)u^c$$

for detail, see ([2], Eq-(6.2)).

By the help of the Euler operator $\vartheta =u{\displaystyle \frac{d}{du}}$, the Touchard polynomials $T_m(u)$ are also defined as follows:

$$T_m(u)=e^{-u}{\vartheta }^m\left\{e^u\right\}$$

where ${\vartheta }^m=\vartheta {\vartheta }^{m-1}$ [4].

Here, we note that the Touchard polynomials have only real roots. These roots are all negative except a simple root at $x=0$ (cf. [1], see also [5,8,9,10]).

Observe that for $x\in {\mathbb{N}}_0$ and $t\in \mathbb{R}$, generating function in (1) is reduced to the moment generating function for the Poisson distribution for suitable values of x, for detail about the Poisson distribution see [11].

When we look at the history of the T(x) polynomials, we see that they were first studied in the literature by Touchard [6,7] in 1933, followed by Bell in 1934, and then by Bromwich [12] in his book on page 95. In addition to the arithmetic properties of the polynomials $T_n(x)$, other important formulas were also given by Carlitz [13]. It can be seen in the literature that combinatorial applications of Tochard or Bell polynomials were given by Riordan [14] and Andrews [15], and their relations with probability distribution functions were also discussed by other researchers.

The Apostol–Bernoulli numbers are defined by

$$F_{ab}(t,\beta )={\displaystyle \frac{t}{\beta e^t-1}}=\sum _{n=0}^{\infty }{\mathcal{B}}_n(\beta ){\displaystyle \frac{t^n}{n!}}$$

(3)

where $t<2\pi$, $\beta =1$ and $t<|log\beta |$, when $\beta \ne 1$ (cf. [16]).

Using (3), we have

$${\mathcal{B}}_0(\beta )=0,$$

$${\mathcal{B}}_1(\beta )={\displaystyle \frac{1}{\beta -1}},$$

$${\mathcal{B}}_2(\beta )={\displaystyle \frac{-2\beta }{{\left(\beta -1\right)}^2}},$$

thus

$${\mathcal{B}}_n(\beta )=\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\mathcal{B}}_j(\beta )$$

(cf. [10,16,17]).

The Apostol–Bernoulli numbers of negative order are defined by

$$F_{ab}(t,\beta ;h)={\left({\displaystyle \frac{\beta e^t-1}{t}}\right)}^h=\sum _{n=0}^{\infty }{\mathcal{B}}_n^{(-h)}(\beta ){\displaystyle \frac{t^n}{n!}}$$

(4)

where $\beta \in \mathbb{C},\mathbb{R}$ and $h\in {\mathbb{Z}}^{+}$ (cf. [18]).

The Apostol–Bernoulli polynomials of negative order are defined by

$$e^{tx}{F}_{ab}(t,\beta ;h)=\sum _{n=0}^{\infty }{\mathcal{B}}_n^{(-h)}(x;\beta ){\displaystyle \frac{t^n}{n!}}$$

(5)

(cf. [18]).

By using (4) and (5), we have the following known formula:

$${\mathcal{B}}_n^{(-h)}(x;\beta )=\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)x^j{\mathcal{B}}_{n-j}^{(-h)}(\beta )$$

(6)

(cf. [10,16]).

Here, we note that

$$F_{ab}(t,\beta )=F_{ab}(t,\beta ;-1)$$

which corresponds to the generating function for the Apostol–Bernoulli polynomials.

The Bernstein polynomials $B_k^n(x)$ are defined by the following generating function, which have been studied by many authors:

$${\mathcal{F}}_B(t,x;k)={\displaystyle \frac{{(tx)}^k}{k!}}{e}^{(1-x)t}=\sum _{n=0}^{\infty }{B}_k^n(x){\displaystyle \frac{t^n}{n!}},$$

(7)

so that

$$B_k^n(x)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)x^k{(1-x)}^{n-k},$$

where $0\le k\le n$, $n\in {\mathbb{N}}_0$ (cf. [19]; see also [20,21,22], the references cited therein).

The beta polynomials of degree n are defined by

$${\mathcal{B}}_{n,k}(x)=x^k{(1+x)}^{n-k},$$

where $0\le k\le n$ (cf. [23]).

The $\lambda$-Stirling numbers of the second kind, $S_2(m,n;\lambda )$ are defined by the following generating function:

$$F_{AS}(t,n;\lambda )={\displaystyle \frac{{(\lambda e^t-1)}^n}{n!}}=\sum _{m=0}^{\infty }{S}_2(m,n;\lambda ){\displaystyle \frac{t^m}{m!}},$$

(8)

where $n\in {\mathbf{N}}_{\mathbf{0}}$ and $\lambda \in \mathbf{C}$.

Applying finite binomial sum to $F_{AS}(t,n;\lambda )$, we have

$$S_2(m,n;\lambda )={\displaystyle \frac{1}{n!}}\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){(-1)}^{n-j}{\lambda }^j{j}^m$$

(9)

(cf. [9,10,17,24]).

Here, we note that

$$F_{AS}(t,n;1)=F_S(t,n).$$

The second author defined the following combinatorial numbers $y_1(m,h;\beta )$:

$${\displaystyle \frac{{(\beta e^t+1)}^h}{h!}}=\sum _{m=0}^{\infty }{y}_1(m,h;\beta ){\displaystyle \frac{t^m}{m!}}$$

(10)

(cf. [25]).

The generating function of polynomials $Y_k^n(x)$ is given by

$$e^{-t}{\mathcal{F}}_B(t,x;k)=\sum _{n=0}^{\infty }{Y}_k^n(x){\displaystyle \frac{t^n}{n!}},$$

where $n\in {\mathbb{N}}_0$, $0\le k\le n$ (cf. [26]).

The polynomials $Y_k^n(x)$ can be written in terms of the Bernstein polynomials. Many properties of these polynomials and their applications related to the Bezier curves were given in [26]. That is, the Bernstein type polynomials $Y_k^n(x)$ are defined by the following equation:

$$Y_k^n(x)=\sum _{j=0}^n{(-1)}^{n-j}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)B_k^j(x),$$

(11)

where $n\in {\mathbb{N}}_0$ and $0\le k\le n$ (cf. [26]).

By combining the following symmetry property of the basis function $B_k^j(x)$:

$$B_k^j(x)=B_{j-k}^j(1-x),$$

Equation (11) can also be written as follows:

$$Y_k^n(x)=\sum _{j=0}^n{(-1)}^{n-j}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)B_{j-k}^j(1-x).$$

(12)

The second author ([18]) defined the following operator:

$$Y_{\lambda ,\beta }[f;a,b](x)=\lambda f(x+a)+\beta f(x+b)$$

(13)

where $a,b$ are real parameters and $\lambda ,\beta$ are real or complex parameters.

By applying the above operator k-times to the function f, the following equation is obtained:

$$Y_{\lambda ,\beta }^k[f;a,b](x)=\sum _{j=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\lambda }^{k-j}{\beta }^{j}f(x+jb+(k-j)a)$$

(14)

(cf. [18]).

The operator $Y_{\lambda ,\beta }[f;a,b](x)$ is a generalization of many known operators involving shift operator, forward difference operator, backward difference operator etc. For detail see [18].

We now summarize the following content of this paper:

In Section 2, we construct generating function for the generalization of the Touchard polynomials. We give some properties of this generating function. We also give a relation between for this generating function, moment generating function for probability distribution, and new polynomials involving the Bernstein polynomials and the beta polynomials.

In Section 3, by using generating functions with their functional equations method, we derive many new identities, finite and infinite sums including exponential function, $\lambda$-Stirling numbers, the Array polynomials, the Apostol type Bernoulli and Euler numbers of negative order, the combinatorial numbers etc.

We conclude this paper with the Section 4.

2. Generalization of the Touchard Polynomials and Their Generating Function

In this section, we construct a new generating function for the generalization of the Touchard polynomials. By using this generating function, we give some formulas for these polynomials, the Bernstein polynomials, and the beta polynomials. By applying the Euler operator $\vartheta =\lambda {\displaystyle \frac{d}{d\lambda }}$ and ${\displaystyle \frac{d}{d\lambda }}$ to this generating function and polynomials, we derive the following novel formulas for the generalization of the Touchard polynomials.

We define a generalization of the Touchard polynomials (generalization exponential functions) in terms of the $\lambda$-Stirling numbers of the second kind as follows:

$$T_m(u;\lambda )=\sum _{n=0}^{\infty }{u}^n{S}_2(m,n;\lambda ),$$

(15)

where $m\in \mathbf{N}=\{1,2,\dots \}$, $x\in \mathbf{R}$ and $\lambda \in \mathbf{C}$ (or $\in \mathbf{R}$) with $\lambda \ne 1$.

When $\lambda =1$,

$$T_m(u)=T_m(u;1).$$

It is easy to rewrite Equation (15) as follows:

$$T_m(u;\lambda )=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n}{n!}}\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){(-1)}^{n-j}{\lambda }^j{j}^m.$$

(16)

Substituting $\lambda =1$ into (9) and using $S_2(m,n;1)=0$ if $n>m$, we get the Touchard polynomials, given in (2).

Theorem 1.

Let $m\in \mathbb{N}$. Then, we have

$$T_m(u;\lambda )=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n}{n!}}\sum _{j=0}^m{\lambda }^j{S}_2(m,j){\displaystyle \frac{d^j}{d{\lambda }^j}}{(\lambda -1)}^n.$$

(17)

Proof. 

By applying the operator ${\vartheta }^m$ to the (9), we have

$$S_2(m,n;\lambda )={\displaystyle \frac{1}{n!}}{\vartheta }^m\left\{{\left(\lambda -1\right)}^n\right\}.$$

(18)

If we substitute the Equation (18) into the Equation (15), we get

$$T_m(u;\lambda )=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n}{n!}}{\vartheta }^m\left\{{\left(\lambda -1\right)}^n\right\}.$$

(19)

By applying the following Euler derivative operator formula, given in [27]:

$${\vartheta }^m\{f(\lambda )\}=\sum _{j=0}^m{S}_2(m,j){\lambda }^j{\displaystyle \frac{d^j}{d{\lambda }^j}}f(\lambda )$$

to (19), we arrive at the desired result. □

Similarly, for $1\le m$, other values of the function $T_m(u;\lambda )$ are also found with the help of the formulas we gave above.

We now construct a generating function for generalization of the Touchard polynomials

$$\begin{array}{ccc}\hfill F(t,u;\lambda )& =& e^{u(\lambda e^t-1)}\hfill \\ & =& \sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}},\hfill \end{array}$$

(20)

where $m\in \mathbf{N}=\{1,2,\dots \}$, $x\in \mathbf{R}$, $\lambda \in \mathbf{C}$ (or $\in \mathbf{R}$) with $\lambda \ne 1$ and $T_m(u;\lambda )$ is generalization exponential functions.

Here we note that (15) and (17) can also be derived from (20). Consequently, using (20), we get

$$\sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n{(\lambda e^t-1)}^n}{n!}}.$$

(21)

Combining the above equation with (8), we obtain

$$\begin{array}{c}\hfill \sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{n=0}^{\infty }{u}^n\sum _{m=0}^{\infty }{S}_2(m,n;\lambda ){\displaystyle \frac{t^m}{m!}}.\end{array}$$

Comparing coefficients on the both sides of the above equation yields not only (15), but also (16).

2.1. Computational Formulas Involving Polynomials $T_m(u;\lambda )$ and $T_m(u)$

Here, by using the higher order Euler derivative operator and ${\displaystyle \frac{d}{dt}}{|}_{t=0}$, we derive some novel computational formulas involving polynomials $T_m(u;\lambda )$ and $T_m(u)$.

By using (16) and (19), we get the following novel formula for the function $T_m(u;\lambda )$:

Theorem 2.

Let $m\in \mathbb{N}$. Then, we have

$$T_m(u;\lambda )={\vartheta }^m\left\{e^{u(\lambda -1)}\right\}$$

(22)

where $\lambda \ne 1$.

We observe that few values of the function $T_m(u;\lambda )$ can be computed as follows:

$$\begin{array}{ccc}\hfill T_1(u;\lambda )& =& \lambda {\displaystyle \frac{d}{d\lambda }}\left\{\sum _{n=0}^{\infty }{u}^n{(\lambda -1)}^n{\displaystyle \frac{1}{n!}}\right\}\hfill \\ & =& \lambda u{e}^{u(\lambda -1)},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill T_2(u;\lambda )& =& \lambda {\displaystyle \frac{d}{d\lambda }}\left\{\lambda {\displaystyle \frac{d}{d\lambda }}\sum _{n=0}^{\infty }{u}^n{(\lambda -1)}^n{\displaystyle \frac{1}{n!}}\right\}\hfill \\ & =& \lambda {\displaystyle \frac{d}{d\lambda }}\left\{\lambda u{e}^{u(\lambda -1)}\right\}\hfill \\ & =& \left(\lambda u+{\lambda }^2{u}^2\right)e^{u(\lambda -1)},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill T_3(u;\lambda )& =& \lambda {\displaystyle \frac{d}{d\lambda }}\left\{\lambda {\displaystyle \frac{d}{d\lambda }}\left\{\lambda {\displaystyle \frac{d}{d\lambda }}\sum _{n=0}^{\infty }{u}^n{(\lambda -1)}^n{\displaystyle \frac{1}{n!}}\right\}\right\}\hfill \\ & =& \lambda {\displaystyle \frac{d}{d\lambda }}\left\{\lambda {\displaystyle \frac{d}{d\lambda }}\lambda u{e}^{u(\lambda -1)}\right\}\hfill \\ & =& \lambda {\displaystyle \frac{d}{d\lambda }}\left\{\left(\lambda u+{\lambda }^2{u}^2\right)e^{u(\lambda -1)}\right\}\hfill \\ & =& \left(\lambda u+2{\lambda }^2{u}^2+{\lambda }^2{u}^2+{\lambda }^3{u}^3\right)e^{u(\lambda -1)},\hfill \end{array}$$

and so on. It is easy to see that for $\lambda =1$, the above functions reduce to the classical Touchard polynomials. Therefore, using (22), we get the following corollary:

Corollary 1.

Let $m\in \mathbb{N}$. Then, we have

$$T_m(u)={\vartheta }^m\left\{e^{u(\lambda -1)}\right\}{|}_{\lambda =1}.$$

(23)

Applying derivative operator ${\displaystyle \frac{d}{dt}}{|}_{t=0}$ to (20), we get the following corollary:

Corollary 2.

Let $m\in \mathbb{N}$. Then, we have

$$T_m(u)={\displaystyle \frac{d^m}{d{t}^m}}\left\{F(t,u;\lambda \right\}{|}_{t=0}.$$

(24)

2.2. Relation Between Generating Function for Generalization of the Touchard Polynomials and Moment Generating Function for Probability Distribution

Here, we give relation between generating function in (20) and moment generating function for probability distribution and the Poisson distribution.

Although the generating function in (1) is the moment generating function for the Poisson distribution, generating function in (20) is not associated with any moment generating function for probability distribution.

We now explain this claim as follows: assuming that (20) is a moment generating function for any probability distribution, which is also denoted by $P_x(\lambda ,u)$. By using definition of moment generating function or generating function for the distribution function yields

$$e^{u(\lambda z-1)}=\sum _{x=0}^{\infty }{z}^x{P}_x(\lambda ,u),$$

where $\lambda >0$ and $z=e^t$. Therefore, using Taylor series of $e^{\lambda uz}$, we have

$$e^{-u}\sum _{x=0}^{\infty }{\displaystyle \frac{{(uz\lambda )}^x}{x!}}=\sum _{x=0}^{\infty }{z}^x{P}_x(\lambda ,u).$$

After some calculations, we get the following discrete probability distribution type function

$$\begin{array}{c}\hfill P_x(\lambda ,u)={\displaystyle \frac{u^x{\lambda }^x}{x!}}{e}^{-u}\end{array}$$

(25)

where $u>0,\lambda >0$ and $x\in {\mathbb{N}}_0$.

Due to the factor ${\lambda }^x$, $P_x(\lambda ,u)$ is neither discrete probability distribution on ${\mathbb{N}}_0$ nor continuous probability distribution on the interval $[0,\infty )$.

If we modify (25), then we get the following well-known Poisson distribution with $x\in {\mathbb{N}}_0$ and parameter $u>0$:

$${\lambda }^{-x}{P}_x(\lambda ,u).$$

(26)

For this distribution see also (cf. [28,29]).

Putting $u=u\lambda$ in (26), we showed that ${\lambda }^{-x}{P}_x(\lambda ,u\lambda )$ is a generating function for the polynomials $Y_x^n(u)$ (cf. [26]).

By combining (25) with (7), we get

$$\begin{array}{c}\hfill P_x(\lambda ,u)={\displaystyle \frac{u^x{\lambda }^x}{x!}}{e}^{-u}=\sum _{n=0}^{\infty }{B}_x^n(\lambda ){\displaystyle \frac{u^n}{n!}}\sum _{k=0}^{\infty }{(-\lambda )}^x{\displaystyle \frac{u^k}{k!}}.\end{array}$$

Therefore,

$$P_x(\lambda ,u)=\sum _{n=0}^{\infty }\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)B_x^k(\lambda ){(-\lambda )}^{n-k}{\displaystyle \frac{u^n}{n!}}.$$

Consequently, we show that $P_x(\lambda ,u)$ is a generating function for the following new polynomials in terms of the Bernstein polynomials:

$$Q_n(\lambda ,x)=\sum _{k=0}^n{(-1)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\lambda }^{n-k}{B}_x^k(\lambda ).$$

(27)

The polynomial $Q_n(\lambda ,x)$ can be written as in terms of the beta polynomials. That is,

$$Q_n(\lambda ,x)=\sum _{k=0}^n{(-1)}^{n-k+x}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{k}{x}}\right){\lambda }^{n-k}{\mathcal{B}}_{k,x}(-\lambda ),$$

(28)

where m and n are nonnegative integers with $m\le n$.

Observe that the beta polynomials, ${\mathcal{B}}_{n,m}(\lambda )$, are different from the Swiss-Knife polynomials with integers coefficients. The Swiss-Knife polynomials are well known to have important relationships with many polynomial and number families, see A153641-OEIS and A162660-OEIS in the online encyclopedia of integer sequences. The most important of these families are the Bernoulli polynomials, Euler polynomials, and the Genocchi number and polynomials see A154344-OEIS.

3. Identities Involving Generalization of the Touchard Polynomials, Finite and Infinite Sums and Also Other Special Numbers and Polynomials

In this section, by using generating function method, we derive many new identities involving generalization of the Touchard polynomials, $\lambda$-Stirling numbers, the Array polynomials, the Apostol type Bernoulli and Euler numbers of negative order, the combinatorial numbers etc. We also give some formulas for finite and infinite sums involving generalization of the Touchard polynomials.

Combining (17) with the following formula

$$\begin{array}{c}\hfill Y_{\lambda ,-1}^n[u^m;1,0](x){|}_{u=0}=\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\lambda }^{n-j}{(-1)}^j{(n-j)}^m\end{array}$$

(29)

(cf. [18]), we get the following theorem:

Theorem 3.

Let m be any nonnegative integer. Then, we have

$$T_m(u;\lambda )=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n}{n!}}{Y}_{\lambda ,-1}^n[u^m;1,0]{|}_{u=0}.$$

(30)

Here we note that $Y_{\lambda ,-1}^n[u^m;1,0]$ is related to the Array polynomials. That is

$$S_n^m(u)=Y_{\lambda ,-1}^n[u^m;1,0].$$

Theorem 4.

Let m be nonnegative integers. Then, we have

$$\begin{array}{c}\hfill T_m(u;\lambda )=\sum _{n=0}^{\infty }{(-1)}^n{u}^n{y}_1(m,n;-\lambda ).\end{array}$$

(31)

Proof. 

Combining (21) with (10), we get

$$\sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{n=0}^{\infty }{(-1)}^n{u}^n\sum _{m=0}^{\infty }{y}_1(m,n;-\lambda ){\displaystyle \frac{t^m}{m!}}.$$

(32)

Comparing coefficient ${\displaystyle \frac{t^m}{m!}}$ on the both sides of the above equation, we yield the desired result. □

Combining (31) with Equation (28) in [25], after some algebraic calculations, we arrive at the following theorem:

Theorem 5.

Let m be nonnegative integers. Then, we have

$$\begin{array}{c}\hfill T_m(u;\lambda )=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-1)}^n{(2u)}^n{\mathcal{E}}_m^{(-n)}(-\lambda )}{n!}}\end{array}$$

(33)

where ${\mathcal{E}}_m^{(-n)}(-\lambda )$ denotes the Apostol–Euler numbers of negative order.

Theorem 6.

Let m be nonnegative integers. Then, we have

$$T_m(u;\lambda )=\sum _{n=0}^{\infty }{\displaystyle \frac{m(m-1)(m-2)\dots (m-n+1)}{n!}}{u}^n{\mathcal{B}}_{m-n}^{(-n)}(\lambda ).$$

(34)

Proof. 

Combining (21) with (4), we get

$$\sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n}{n!}}\sum _{m=0}^{\infty }{\mathcal{B}}_m^{(-n)}(\lambda ){\displaystyle \frac{t^{m+n}}{m!}}.$$

Therefore

$$\sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{n=0}^{\infty }{\displaystyle \frac{u^n}{n!}}\sum _{m=0}^{\infty }m(m-1)(m-2)\dots (m-n+1){\mathcal{B}}_{m-n}^{(-n)}(\lambda ){\displaystyle \frac{t^m}{m!}}.$$

Comparing coefficient ${\displaystyle \frac{t^m}{m!}}$ on the both sides of the above equation, we yield the desired result. □

Remark 1.

For $n\in \mathbb{N}$, combining (5) with (8) yields a relation between the numbers ${\mathcal{B}}_m^{(-n)}(0;\lambda )$ and $S_2(m,n;\lambda )$. That is

$$\sum _{m=0}^{\infty }n!S_2(m,n;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{m=0}^{\infty }{\mathcal{B}}_m^{(-n)}(0;\lambda ){\displaystyle \frac{t^{m+n}}{m!}}.$$

Therefore

$$S_2(m,n;\lambda )=\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\mathcal{B}}_{m-n}^{(-n)}(0;\lambda ).$$

Substituting the above formula into (15), we also arrive at (34).

Theorem 7.

Let m and k be nonnegative integers. Then, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d^k}{d{u}^k}}\{T_m(u;\lambda )\}=k!\sum _{v=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{v}}\right)S_2(v,k;\lambda )T_{m-v}(u;\lambda ).\end{array}$$

(35)

Proof. 

By applying derivative operator ${\displaystyle \frac{d^k}{d{u}^k}}$ to (20), we get

$$\begin{array}{c}\hfill {(\lambda e^t-1)}^k\sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{m=0}^{\infty }{\displaystyle \frac{d^k}{d{u}^k}}\{T_m(u;\lambda )\}{\displaystyle \frac{t^m}{m!}}.\end{array}$$

Combining the left-hand side of the above equation with (8), we obtain

$$\begin{array}{c}\hfill k!\sum _{m=0}^{\infty }{S}_2(m,k;\lambda ){\displaystyle \frac{t^m}{m!}}\sum _{m=0}^{\infty }{T}_m(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{m=0}^{\infty }{\displaystyle \frac{d^k}{d{u}^k}}\{T_m(u;\lambda )\}{\displaystyle \frac{t^m}{m!}}.\end{array}$$

By applying the Cauchy product rule to the left-hand side of the above equation yields

$$\begin{array}{c}\hfill k!\sum _{m=0}^{\infty }\sum _{v=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{v}}\right)S_2(v,k;\lambda )T_{m-v}(u;\lambda ){\displaystyle \frac{t^m}{m!}}=\sum _{m=0}^{\infty }{\displaystyle \frac{d^k}{d{u}^k}}\{T_m(u;\lambda )\}{\displaystyle \frac{t^m}{m!}}.\end{array}$$

Comparing coefficients ${\displaystyle \frac{t^m}{m!}}$ on the both sides of the above equation yields desired result. □

4. Conclusions

We gave generating functions for the Touchard polynomials. We also defined these polynomials with the aid of the Euler derivative operator. We showed that when $\lambda =1$, generating function for the generalization of the Touchard polynomials reduced to that of Touchard polynomials. We also constructed generating function for generalization of the Touchard polynomials. We gave some properties of this function. By using this function with the aid of probability distribution, we also gave generating function for a new class of polynomials in terms of the Bernstein polynomials and the beta polynomials. With the aid of this function, we derived many new formulas associated with $\lambda$-Stirling numbers, the Array polynomials, the Apostol type Bernoulli and Euler numbers of negative order, the combinatorial numbers etc. Furthermore, by applying derivative operator to the generating function for generalization of the Touchard polynomials, we also derived higher order derivative formula for the generalization of the Touchard polynomials.