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On a class of generalized q-Bernoulli and q-Euler polynomials
Advances in Difference Equations volume 2013, Article number: 115 (2013)
The main purpose of this paper is to introduce and investigate a new class of generalized q-Bernoulli and q-Euler polynomials. The q-analogues of well-known formulas are derived. A generalization of the Srivastava-Pintér addition theorem is obtained.
Throughout this paper, we always make use of the following notation: ℕ denotes the set of natural numbers, denotes the set of nonnegative integers, ℝ denotes the set of real numbers, ℂ denotes the set of complex numbers.
The q-numbers and q-factorial are defined by
respectively. The q-polynomial coefficient is defined by
The q-analogue of the function is defined by
The q-binomial formula is known as
In the standard approach to the q-calculus, two exponential functions are used:
From this form, we easily see that . Moreover,
where is defined by
The above q-standard notation can be found in .
Carlitz firstly extended the classical Bernoulli and Euler numbers and polynomials, introducing them as q-Bernoulli and q-Euler numbers and polynomials [2–4]. There are numerous recent investigations on this subject by, among many other authors, Cenki et al. [5–7], Choi et al.  and , Kim et al. [10–13], Ozden and Simsek , Ryoo et al. , Simsek [16, 17] and , and Luo and Srivastava , Srivastava et al. , Mahmudov [21, 22].
Motivated by the generalizations in (1) of the classical Bernoulli and Euler polynomials, we introduce and investigate here the so-called generalized two-dimensional q-Bernoulli and q-Euler polynomials, which are defined as follows.
Definition 1 Let , , . The generalized two-dimensional q-Bernoulli polynomials are defined, in a suitable neighborhood of , by means of the generating function
Definition 2 Let , , . The generalized two-dimensional q-Euler polynomials are defined, in a suitable neighborhood of , by means of the generating functions
It is obvious that
In fact Definitions 1 and 2 define two different types and of the generalized q-Bernoulli polynomials and two different types and of the generalized q-Euler polynomials. Both polynomials and ( and ) coincide with the classical higher-order generalized Bernoulli polynomials (Euler polynomials) in the limiting case .
2 Preliminaries and lemmas
In this section we provide some basic formulas for the generalized q-Bernoulli and q-Euler polynomials to obtain the main results of this paper in the next section. The following result is a q-analogue of the addition theorem for the classical Bernoulli and Euler polynomials.
Lemma 3 For all we have
In particular, setting and in (3) and (4), we get the following formulae for the generalized q-Bernoulli and q-Euler polynomials, respectively,
Setting and in (3) and (4), we get, respectively,
Clearly, (5) and (6) are the generalization of q-analogues of
Lemma 4 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:
Lemma 5 We have
Lemma 6 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:
Proof We prove only (7). The proof is based on the following equality:
Here we used the following relation:
Corollary 7 Taking , we have
Lemma 8 The generalized q-Bernoulli polynomials satisfy the following relations:
Remark 9 Notice taking limit in (8) as , we get
It is a correct form of formula (2.7) from  for .
Lemma 10 We have
From Lemma 10 we obtain the list of generalized q-Bernoulli polynomials as follows
3 Explicit relationship between the q-Bernoulli and q-Euler polynomials
In this section, we give some generalizations of the Srivastava-Pintér addition theorem. We also obtain new formulae and their some special cases below.
Theorem 11 The relationships
hold true between the generalized q-Bernoulli polynomials and q-Euler polynomials.
Proof First we prove (9). Using the identity
It is clear that
On the other hand,
Next we prove (10). Using the identity
It is clear that
On the other hand,
Next we discuss some special cases of Theorem 11.
Theorem 12 The relationship
holds true between the generalized q-Bernoulli polynomials and the q-Euler polynomials.
Remark 13 Taking in Theorem 12, we obtain the Srivastava-Pintér addition theorem for the generalized Bernoulli and Euler polynomials.
Notice that the Srivastava-Pintér addition theorem for the generalized Apostol-Bernoulli polynomials and the Apostol-Euler polynomials was given in . The formula (11) is a correct version of Theorem 3  for .
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Dedicated to Professor Hari M Srivastava.
The authors would like to thank the reviewers for their valuable comments and helpful suggestions that improved the note’s quality.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.