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 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}}^{+}, when s,z\in \mathbb{C} (); \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 . 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

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

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. □