Research

# On the generalized Apostol-type Frobenius-Euler polynomials

Burak Kurt1 and Yilmaz Simsek2*

Author Affiliations

1 Department of Mathematics, Faculty of Education, Akdeniz University, Antalya, 07058, Turkey

2 Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, 07058, Turkey

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Advances in Difference Equations 2013, 2013:1  doi:10.1186/1687-1847-2013-1

 Received: 8 November 2012 Accepted: 13 December 2012 Published: 4 January 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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.

##### Keywords:
Frobenius-Euler polynomials; Hermite-based Frobenius-Euler polynomials; Hermite-based Apostol-Euler polynomials; Apostol-Euler polynomials; Hurwitz-Lerch zeta function

### 1 Introduction, definitions and notations

Throughout this presentation, we use the following standard notions: , , . 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, and

where , .

The classical Frobenius-Euler polynomial of order α is defined by means of the following generating function:

(1)

where u is an algebraic number and .

Observe that , which denotes the Frobenius-Euler polynomials and , which denotes the Frobenius-Euler numbers of order α. , which denotes the Euler polynomials (cf.[1-24]).

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

Let , , . The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function:

(2)

Remark 1.2 If we set and in (2), we get

(3)

where 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 2.1Let. Each of the following relationships holds true:

(4)

(5)

(6)

and

(7)

Proof of (6) From (2),

(8)

Therefore,

Thus, by using the Cauchy product in (8) and then equating the coefficients of 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

where is replaced by .

Theorem 2.2Let. Then we have

Proof By using (2), we get

By equating the coefficients of on both sides of the resulting equation, we obtain the desired result. □

Theorem 2.3The following relationship holds true:

(9)

Proof We set

From the above equation, we see that

Therefore,

Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □

Remark 2.4 By substituting , , into Theorem 2.3, we get Carlitz’s results (for details, see [[1], Eq. 2.19]) as follows:

We give the following generating function of the polynomials :

(10)

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

If we substitute and into (10), we see that

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

(11)

Proof Substituting for into (2) and taking derivative with respect to t, we obtain

Using (10), we have

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

Theorem 2.6Letand. Then we have

(12)

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

By using (2), we get

Therefore,

Comparing the coefficients of 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 is given as follows:

(, , , when (); and when ). It is obvious that

(13)

and

where 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 3.1Let. We have

(14)

where

Proof From (2), we have

Therefore,

Comparing the coefficients of on both sides of the above equation, we have arrive at (14). □

Remark 3.2 By substituting , into (14), we have

where

Remark 3.3 The function 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 for , .

### 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 by means of the following generating function:

(15)

The generating function for these polynomials is given by

(16)

(cf.[17]).

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

Let with , and . Then the generalized Bernoulli polynomials of order are defined by means of the following generating functions:

(17)

where

We note that and also , which denotes the Apostol-Bernoulli polynomials (cf.[1-24]).

Theorem 4.1Letvbe an integer. Then we have

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

Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □

Corollary 4.2

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

Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

### Acknowledgements

Dedicated to Professor Hari M. Srivastava.

All authors are partially supported by Research Project Offices Akdeniz Universities.

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