Abstract and Applied Analysis

Volume 2013, Article ID 293532, 6 pages

http://dx.doi.org/10.1155/2013/293532

## Some Identities on the Generalized *q*-Bernoulli, *q*-Euler, and *q*-Genocchi Polynomials

^{1}National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon 305-811, Republic of Korea^{2}Department of Mathematics, Akdeniz University, 07058 Antalya, Turkey

Received 13 September 2013; Accepted 12 November 2013

Academic Editor: Junesang Choi

Copyright © 2013 Daeyeoul Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Mahmudov (2012, 2013) introduced and investigated some -extensions of the -Bernoulli polynomials of order , the -Euler polynomials of order , and the -Genocchi polynomials of order . In this paper, we give some identities for , , and and the recurrence relations between these polynomials. This is an analogous result to the -extension of the Srivastava-Pintér addition theorem in Mahmudov (2013).

#### 1. Introduction, Definitions, and Notations

Throughout this paper, we always make use of the following notation: denotes the set of natural numbers and denotes the set of complex numbers. The -numbers and -factorial are defined by respectively, where , , and . The -binomial coefficient is defined by where . The -analogue of the function is defined by The -binomial formula is known as

The -exponential functions are given by From these forms, we easily see that . Moreover, and , where is defined by

The previous -standard notation can be found in [1, 2]. Carlitz firstly extended the classical Bernoulli numbers and polynomials and Euler numbers and polynomials [3, 4]. There are numerous recent investigations on this subject by many other authors. Among them are Cenkci et al. [5, 6], Choi et al. [1], Cheon [7], Kim [8], Kurt [9], Kurt [10], Luo and Srivastava [11–13], Srivastava et al. [14, 15], Natalini and Bernardini [16], Tremblay et al. [17, 18], Gaboury and Kurt [19], Mahmudov [2, 20, 21], Araci et al. [22], and Kupershmidt [23].

Mahmudov defined and studied the properties of the following generalized -Bernoulli polynomials of order and -Euler polynomials of order as follows [2].

Let , , and . The -Bernoulli numbers and polynomials in and of order are defined by means of the generating functions: The -Euler numbers and polynomials in and of order are defined by means of the generating functions:

The -Genocchi numbers and polynomials in and of order are defined by means of the generating functions:

The familiar -Stirling numbers of the second kind are defined by

It is obvious that From (8) and (10), it is easy to check that

In this work, we give some identities for the -Bernoulli polynomials. Also, we give some relations between the -Bernoulli polynomials and -Euler polynomials and the -Genocchi polynomials and -Bernoulli polynomials. Furthermore, we give a different form of the analogue of the Srivastava-Pintér addition theorem. More precisely, we prove the following theorems.

Theorem 1. *There are the following relations between the -Bernoulli polynomials and -Stirling numbers of the second kind:
**
where , , and .*

Theorem 2. *The -Stirling numbers of the second kind satisfy the following relations:
**
where , , and .*

Theorem 3. *The generalized -Euler polynomials satisfy the following relation:
**
where , , and .*

Theorem 4. *The polynomials and satisfy the following difference relationships:
**
where , , and .*

Theorem 5. *There is the following relation between the generalized -Euler polynomials and generalized -Bernoulli polynomials:
**
where , , and .*

#### 2. Proof of the Theorems

Lemma 6. *The generalized -Bernoulli polynomials, -Euler polynomials, and -Genocchi polynomials satisfy the following relations:
*

*Proof. *The proof of this lemma can be found from (7)–(12).

*Proof of Theorem 1. *By (8) and (13) we have
Equating the coefficients of , we obtain (16).

Similarly, we have (17).

*Proof of Theorem 2. *Combining (10) and (13), we obtain

Comparing the coefficients of , we find (18). Similarly, we have (19).

*Proof of Theorem 3. *It is obvious that
We write it as
Using the Cauchy product and comparing the coefficients of , we have

Finally, we consider the interesting relationships between the -Bernoulli polynomials and -Genocchi polynomials and the -Euler polynomials and -Bernoulli polynomials. These relations are -analogues to the Srivastava-Pintér addition theorems.

*Proof of Theorem 4. *It follows immediately that
Equating the coefficients of , we have (21).

In a similar fashion, (12) yields
Comparing the coefficients of , we have (22).

*Proof of Theorem 5. *By (10), we write
By equating the coefficients of , we get the theorem.

*Remark 7. *There are many different relationships which are analogues to the Srivastava-Pintér addition theorems at these polynomials.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

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