On the generalized Apostol-type Frobenius-Euler polynomials

Abstract

The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz’s results which are associated with Frobenius-Euler polynomials.

MSC:05A10, 11B65, 28B99, 11B68.

1 Introduction, definitions and notations

Throughout this presentation, we use the following standard notions: $\mathbb{N}=\{1,2,\dots \}$, ${\mathbb{N}}_0=\{0,1,2,\dots \}=\mathbb{N}\cup \{0\}$, ${\mathbb{Z}}^{-}=\{-1,-2,\dots \}$. Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers. Furthermore, ${(\lambda )}_0=1$ and

$${(\lambda )}_k=\lambda (\lambda +1)(\lambda +2)\cdots (\lambda +k-1),$$

where $k\in \mathbb{N}$, $\lambda \in \mathbb{C}$.

The classical Frobenius-Euler polynomial $H_n^{(\alpha )}(x;u)$ of order α is defined by means of the following generating function:

$${\left(\frac{1-u}{e^t-u}\right)}^{\alpha }{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{H}_n^{(\alpha )}(x;u)\frac{t^n}{n!},$$

(1)

where u is an algebraic number and $\alpha \in \mathbb{Z}$.

Observe that $H_n^{(1)}(x;u)=H_n(x;u)$, which denotes the Frobenius-Euler polynomials and $H_n^{(\alpha )}(0;u)=H_n^{(\alpha )}(u)$, which denotes the Frobenius-Euler numbers of order α. $H_n(x;-1)=E_n(x)$, which denotes the Euler polynomials (cf. [1–24]).

Definition 1.1 (for details, see [16, 17])

Let $a,b,c\in {\mathbb{R}}^{+}$, $a\ne b$, $x\in \mathbb{R}$. The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function:

$${\left(\frac{a^t-u}{\lambda b^t-u}\right)}^{\alpha }{c}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(\alpha )}(x;u;a,b,c;\lambda )\frac{t^n}{n!}.$$

(2)

Remark 1.2 If we set $x=0$ and $\alpha =1$ in (2), we get

$$\frac{a^t-u}{\lambda b^t-u}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(u;a,b,c;\lambda )\frac{t^n}{n!},$$

(3)

where ${\mathcal{H}}_n(u;\lambda ;a,b,c)$ denotes the generalized Apostol-type Frobenius-Euler numbers (cf. [17]).

2 New identities

In this section, we derive many new identities related to the generalized Apostol-type Frobenius-Euler numbers and polynomials of order α.

Theorem 2.1 Let $\alpha ,\beta \in \mathbb{Z}$. Each of the following relationships holds true:

(4)

(5)

(6)

and

$${\mathcal{H}}_n^{(-\alpha )}(x;u^2;a^2,b^2,c^2;{\lambda }^2)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathcal{H}}_k^{(-\alpha )}(x;u;a,b,c;\lambda ){\mathcal{H}}_{n-k}^{(-\alpha )}(x;-u;a,b,c;\lambda ).$$

(7)

Proof of (6) From (2),

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(-\alpha )}(x;u;a,b,c;\lambda )\frac{t^n}{n!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(\alpha )}(y;u;a,b,c;\lambda )\frac{t^n}{n!}=c^{(x+y)t}.$$

(8)

Therefore,

$$\sum _{n=0}^{\mathrm{\infty }}(\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathcal{H}}_{n-k}^{(\alpha )}(y;u;a,b,c;\lambda ){\mathcal{H}}_k^{(-\alpha )}(x;u;a,b,c;\lambda ))\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}{(xlnc)}^n\frac{t^n}{n!}.$$

Thus, by using the Cauchy product in (8) and then equating the coefficients of $\frac{t^n}{n!}$ on both sides of the resulting equation, we obtain the desired result.

The proofs of (4), (5) and (7) are the same as that of (2), thus we omit them. □

Observe that in (6) we have

$${((x+y)lnc)}^n={({\mathcal{H}}^{(\alpha )}(y;u;a,b,c;\lambda )+{\mathcal{H}}^{(-\alpha )}(x;u;a,b,c;\lambda ))}^n,$$

where ${({\mathcal{H}}^{(\alpha )}(y;u;a,b,c;\lambda ))}^n$ is replaced by ${\mathcal{H}}_n^{(\alpha )}(y;u;a,b,c;\lambda )$.

Theorem 2.2 Let $\alpha \in \mathbb{N}$. Then we have

$$\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right){(-u)}^{\alpha -k}{(xlnc+klna)}^n=\sum _{p=0}^n\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{n}{p}\right)\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right){(-u)}^{\alpha -k}{(klnb)}^p{\mathcal{H}}_{n-p}^{(\alpha )}(x;u;a,b,c;\lambda ).$$

Proof By using (2), we get

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right){(-u)}^{\alpha -k}{(xlnc+klna)}^n\right)\frac{t^n}{n!}\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}(\sum _{p=0}^n\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{n}{p}\right)\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right){(-u)}^{\alpha -k}{(klnb)}^p{\mathcal{H}}_{n-p}^{(\alpha )}(x;u;a,b,c;\lambda ))\frac{t^n}{n!}.\hfill \end{array}$$

By equating the coefficients of $\frac{t^n}{n!}$ on both sides of the resulting equation, we obtain the desired result. □

Theorem 2.3 The following relationship holds true:

(9)

Proof We set

$$\begin{array}{c}(2u-1)\frac{a^t-u}{\lambda b^t-u}{c}^{xt}\frac{a^t-(1-u)}{\lambda b^t-(1-u)}{c}^{yt}\hfill \\ =(a^t-u)(a^t-(1-u))c^{(x+y)t}(\frac{1}{\lambda b^t-u}-\frac{1}{\lambda b^t-(1-u)}).\hfill \end{array}$$

From the above equation, we see that

$$\begin{array}{c}(2u-1)(\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(x;u;a,b,c;\lambda )\frac{t^n}{n!})(\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(y;1-u;a,b,c;\lambda )\frac{t^n}{n!})\hfill \\ =(a^t-1+u)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(x+y;u;a,b,c;\lambda )\frac{t^n}{n!}-(a^t-u)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(x+y;1-u;a,b,c;\lambda )\frac{t^n}{n!}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}(2u-1)\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right){\mathcal{H}}_r(x;u;a,b,c;\lambda ){\mathcal{H}}_{n-r}(y;1-u;a,b,c;\lambda )\frac{t^n}{n!}\hfill \\ =(u-1)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(x+y;u;a,b,c;\lambda )\frac{t^n}{n!}+u\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(x+y;1-u;a,b,c;\lambda )\frac{t^n}{n!}\hfill \\ +\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right){(lna)}^{n-r}{\mathcal{H}}_r(x+y;u;a,b,c;\lambda )\frac{t^n}{n!}\hfill \\ -\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right){(lna)}^{n-r}{\mathcal{H}}_r(x+y;1-u;a,b,c;\lambda )\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ on both sides of the above equation, we arrive at the desired result. □

Remark 2.4 By substituting $a=1$, $b=c=e$, $\lambda =1$ into Theorem 2.3, we get Carlitz’s results (for details, see [[1], Eq. 2.19]) as follows:

$$\begin{array}{c}(2u-1)\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)H_r(x;u)H_{n-r}(y;1-u)\hfill \\ =(u-1)H_n(x+y;u)+u{H}_n(x+y;1-u)+H_n(x+y;u)-H_n(x+y;1-u).\hfill \end{array}$$

We give the following generating function of the polynomials $Y_n(x;\lambda ;a)$:

$$\frac{t}{\lambda a^t-1}{a}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{Y}_n(x;\lambda ;a)\frac{t^n}{n!}(a\ge 1)$$

(10)

(cf. [16, 17]). We also note that

$$Y_n(0;\lambda ;a)=Y_n(\lambda ;a).$$

If we substitute $x=0$ and $a=1$ into (10), we see that

$$Y_n(\lambda ;1)=\frac{1}{\lambda -1}.$$

Theorem 2.5 The generalized Apostol-type Frobenius-Euler polynomial holds true as follows:

(11)

Proof Substituting $c=b$ for $\alpha =1$ into (2) and taking derivative with respect to t, we obtain

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n+1}(x;u;a,b,b;\lambda )\frac{t^n}{n!}\hfill \\ =\frac{a^{t}lna}{a^t-u}\frac{a^t-u}{\lambda b^t-u}{b}^{xt}+\frac{lnb\lambda b^t}{a^t-u}{\left(\frac{a^t-u}{\lambda b^t-u}\right)}^2{b}^{xt}+ln\left(b^x\right)\frac{a^t-u}{\lambda b^t-u}{b}^{xt}.\hfill \end{array}$$

Using (10), we have

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n+1}(x;u;a,b,b;\lambda )\frac{t^n}{n!}& =& \frac{ln(a^{\frac{1}{u}})}{t}\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)Y_{n-k}(1;\frac{1}{u};a){\mathcal{H}}_k(x;u;a,b,b;\lambda )\frac{t^n}{n!}\\ +\frac{ln(b^{\frac{\lambda }{u}})}{t}\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)Y_{n-k}(\frac{1}{u};a){\mathcal{H}}_k^{(2)}(x;u;a,b,b;\lambda )\frac{t^n}{n!}\\ +ln\left(b^x\right)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n(x;u;a,b,b;\lambda )\frac{t^n}{n!}.\end{array}$$

Thus, after some elementary calculations, we arrive at (11). □

Theorem 2.6 Let $|u|<1$ and $m\in \mathbb{N}$. Then we have

$${\mathcal{H}}^{(-m)}(u;a,b,c;\lambda )=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathcal{H}}_k^{(-\alpha )}(-x;u;a,b,c;\lambda ){\mathcal{H}}_{n-k}^{(\alpha -m)}(x;u;a,b,c;\lambda ).$$

(12)

Proof In (2), we replace α by −α, then we set

$${\left(\frac{a^t-u}{\lambda b^t-u}\right)}^{-\alpha }{c}^{(-x)t}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(\alpha -m)}(x;u;a,b,c;\lambda )\frac{t^n}{n!}={\left(\frac{a^t-u}{\lambda b^t-u}\right)}^{-m}.$$

By using (2), we get

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(-\alpha )}(-x;u;a,b,c;\lambda )\frac{t^n}{n!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(\alpha -m)}(x;u;a,b,c;\lambda )\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(-m)}(u;a,b,c;\lambda )\frac{t^n}{n!}.$$

Therefore,

$$\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathcal{H}}_k^{(-\alpha )}(-x;u;a,b,c;\lambda ){\mathcal{H}}_{n-k}^{(\alpha -m)}(x;u;a,b,c;\lambda )\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(-m)}(u;a,b,c;\lambda )\frac{t^n}{n!}.$$

Comparing the coefficients of $\frac{t^n}{n!}$ on both sides of the above equation, we arrive at (12). □

3 Interpolation function

In this section, we give a recurrence relation between the generalized Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function. Recently, many authors have studied not only the Hurwitz-Lerch zeta function, but also its generalizations, for example (among others), Srivastava [19], Srivastava and Choi [24] and also Garg et al. [6]. The generalization of the Hurwitz-Lerch zeta function $\mathrm{\Phi }(z,s,a)$ is given as follows:

$${\mathrm{\Phi }}_{\mu ,\nu }^{(\rho ,\sigma )}(z,s,a):=\sum _{n=0}^{\mathrm{\infty }}\frac{{(\mu )}_{\rho n}}{{(\nu )}_{\sigma n}}\frac{z^n}{{(n+a)}^s}$$

($\mu \in \mathbb{C}$, $a,\upsilon \in \mathbb{C}\mathrm{\setminus }{\mathbb{Z}}_0^{-}$, $\rho ,\sigma \in {\mathbb{R}}^{+}$, $\rho <\sigma$ when $s,z\in \mathbb{C}$ ($|z|<1$); $\rho =\sigma$ and $\mathrm{\Re }(s-\mu +\nu )>0$ when $|z|=1$). It is obvious that

$${\mathrm{\Phi }}_{\mu ,1}^{(1,1)}(z,s,a)={\mathrm{\Phi }}_{\mu }^{\ast }(z,s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(\mu )}_n}{n!}\frac{z^n}{{(n+a)}^s}$$

(13)

and

$${\mathrm{\Phi }}_n^{\ast }(z,s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(n)}_n}{n!}\frac{z^n}{{(n+a)}^s}=\mathrm{\Phi }(z,s,a),$$

where $\mathrm{\Phi }(z,s,a)$ denotes the Lerch-Zeta function (cf. [6, 19, 21, 24]).

Relation between the generalized Apostol-type Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function is given as follows.

Theorem 3.1 Let $|\frac{\lambda }{u}|<1$. We have

$${\mathcal{H}}_n^{(\alpha )}(x;u;a,b,c;\lambda )=\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right){(-u)}^{\alpha -k-1}\mathfrak{G}(-n;x,\frac{\lambda }{u};a,b,c;\alpha ,k),$$

(14)

where

$$\mathfrak{G}(s;x,\beta ;a,b,c;\alpha ,j)=\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{m+\alpha -1}{m}\right)\frac{{\beta }^m}{{(xlnc+jlna+mlnb)}^s},|\beta |<1.$$

Proof From (2), we have

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(\alpha )}(x;u;a,b,c;\lambda )\frac{t^n}{n!}=\sum _{j=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{\alpha }{j}\right){(-u)}^{\alpha -j-1}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{m+\alpha -1}{m}\right){\left(\frac{\lambda }{u}\right)}^m{e}^{\alpha (xlnc+klna+mlnb)}.$$

Therefore,

$$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(\alpha )}(x;u;a,b,c;\lambda )\frac{t^n}{n!}\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right){(-u)}^{\alpha -k-1}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{m+\alpha -1}{m}\right){\left(\frac{\lambda }{u}\right)}^m{(xlnc+klna+mlnb)}^n\frac{t^n}{n!}.\hfill \end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ on both sides of the above equation, we have arrive at (14). □

Remark 3.2 By substituting $a=1$, $b=c=e$ into (14), we have

$${\mathcal{H}}_n^{(\alpha )}(x;u;\lambda )=-\frac{{(1-u)}^{\alpha }}{u}\mathfrak{G}(-n;x,\frac{\lambda }{u};1,e,e;\alpha ,1)=-\frac{{(1-u)}^{\alpha }}{u}\mathrm{\Phi }(\frac{\lambda }{u},-n,x),$$

where

$$\mathfrak{G}(-n;x,\frac{\lambda }{u};1,e,e;\alpha ,1)=\mathrm{\Phi }(\frac{\lambda }{u},-n,x).$$

Remark 3.3 The function $\mathfrak{G}(s;x,\beta ;a,b,c;\alpha ,j)$ is an interpolation function of the generalized Apostol-type Frobenius-Euler polynomials of order α at negative integers, which is given by the analytic continuation of the $\mathfrak{G}(s;x,\beta ;a,b,c;\alpha ,j)$ for $s=-n$, $n\in \mathbb{N}$.

4 Relations between Array-type polynomials, Apostol-Bernoulli polynomials and generalized Apostol-type Frobenius-Euler polynomial

In [17], Simsek constructed the generalized λ-Stirling type numbers of the second kind $\mathcal{S}(n,v;a,b;\lambda )$ by means of the following generating function:

$$f_{S,v}(t;a,b;\lambda )=\frac{{(\lambda b^t-a^t)}^v}{v!}=\sum _{n=0}^{\mathrm{\infty }}\mathcal{S}(n,v;a,b;\lambda )\frac{t^n}{n!}.$$

(15)

The generating function for these polynomials ${\mathcal{S}}_v^n(x;a,b;\lambda )$ is given by

$$g_v(x,t;a,b;\lambda )=\frac{1}{v!}{(\lambda b^t-a^t)}^v{b}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{S}}_v^n(x;a,b;\lambda )\frac{t^n}{n!}$$

(16)

(cf. [17]).

The generalized Apostol-Bernoulli polynomials were defined by Srivastava et al. [[22], p.254, Eq. (20)] as follows.

Let $a,b,c\in {\mathbb{R}}^{+}$ with $a\ne b$, $x\in \mathbb{R}$ and $n\in {\mathbb{N}}_0$. Then the generalized Bernoulli polynomials ${\mathfrak{B}}_n^{(\alpha )}(x;\lambda ;a,b,c)$ of order $\alpha \in \mathbb{Z}$ are defined by means of the following generating functions:

$$f_B(x,a,b,c;\lambda ;\alpha )={\left(\frac{t}{\lambda b^t-a^t}\right)}^{\alpha }{c}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathfrak{B}}_n^{(\alpha )}(x;\lambda ;a,b,c)\frac{t^n}{n!},$$

(17)

where

$$|tln\left(\frac{a}{b}\right)+ln\lambda |<2\pi .$$

We note that ${\mathfrak{B}}_n^{(1)}(x;\lambda ;a,b,c)={\mathfrak{B}}_n(x;\lambda ;a,b,c)$ and also ${\mathfrak{B}}_n(x;\lambda ;1,e,e)=B_n(x;\lambda )$, which denotes the Apostol-Bernoulli polynomials (cf. [1–24]).

Theorem 4.1 Let v be an integer. Then we have

$${\mathcal{H}}_{n-v}^{(-\nu )}(x;u;a,b,c;\lambda )=\frac{\nu !}{u^{2\nu }{(n)}_v}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){\mathcal{S}}_v^n(x,1,b;\frac{\lambda }{u})Y_{n-k}^{(\nu )}(\frac{1}{u};a).$$

Proof Replacing c by b in (2) and after some calculations, we have

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_n^{(-v)}(x;u;a,b,b;\lambda )\frac{t^{n+v}}{n!}=\frac{\nu !}{u^{2\nu }}\sum _{n=0}^{\mathrm{\infty }}{S}_{\nu }^n(x,1,b;\frac{\lambda }{u})\frac{t^n}{n!}\sum _{n=0}^{\mathrm{\infty }}{Y}_n^{(\nu )}(\frac{1}{u};a)\frac{t^n}{n!}.$$

Comparing the coefficients of $\frac{t^n}{n!}$ on both sides of the above equation, we arrive at the desired result. □

Corollary 4.2

$${\mathcal{H}}_{n-v}^{(-\nu )}(x;u;a,b,c;\lambda )=\frac{\nu !}{u^{2\nu }{(n)}_{\alpha }}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\mathcal{S}(k,\nu ,1,b;\frac{\lambda }{u}){\mathfrak{B}}_{n-k}(x,a,b;\frac{\lambda }{u}).$$

Proof Replacing c by b in (2) and after some calculations, we have

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n-v}^{(-v)}(x;u;a,b,b;\lambda )\frac{t^{n+v}}{n!}=\frac{\nu !}{u^{2\nu }}\sum _{n=0}^{\mathrm{\infty }}\mathcal{S}(n,\nu ,1,b;\frac{\lambda }{u})\frac{t^n}{n!}\sum _{n=0}^{\mathrm{\infty }}{\mathfrak{B}}_n(x,a,b;\frac{\lambda }{u})\frac{t^n}{n!}.$$

Comparing the coefficients of $\frac{t^n}{n!}$ on both sides of the above equation, we arrive at the desired result. □