, 2013:115

# On a class of generalized q-Bernoulli and q-Euler polynomials

Open Access
Research
Part of the following topical collections:
1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

## Abstract

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.

## Keywords

Nonnegative Integer Correct Form Euler Number Basic Formula Addition Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 1 Introduction

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 [1].

Carlitz firstly extended the classical Bernoulli and Euler numbers and polynomials, introducing them as q-Bernoulli and q-Euler numbers and polynomials [2, 3, 4]. There are numerous recent investigations on this subject by, among many other authors, Cenki et al. [5, 6, 7], Choi et al. [8] and [9], Kim et al. [10, 11, 12, 13], Ozden and Simsek [14], Ryoo et al. [15], Simsek [16, 17] and [18], and Luo and Srivastava [19], Srivastava et al. [20], Mahmudov [21, 22].

Recently, Natalini and Bernardini [23], Bretti et al. [24], Kurt [25, 26], Tremblay et al. [27, 28] studied the properties of the following generalized Bernoulli and Euler polynomials:
(1)

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

where .

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

Here and denote the generalized Bernoulli and Euler polynomials defined in (1). Notice that was introduced by Natalini [23], and was introduced by Kurt [25].

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
(2)
(3)
(4)
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,
(5)
(6)
Clearly, (5) and (6) are the generalization of q-analogues of

respectively.

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:
(7)
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:
(8)

Proof

Indeed,

□

Remark 9 Notice taking limit in (8) as , we get

It is a correct form of formula (2.7) from [27] 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.

We present natural q-extensions of the main results of the papers [29, 30].

Theorem 11 The relationships
(9)
(10)

hold true between the generalized q-Bernoulli polynomials and q-Euler polynomials.

Proof First we prove (9). Using the identity
we have
It is clear that
On the other hand,
Therefore
Next we prove (10). Using the identity
we have
It is clear that
On the other hand,
Therefore

□

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.
(11)

Notice that the Srivastava-Pintér addition theorem for the generalized Apostol-Bernoulli polynomials and the Apostol-Euler polynomials was given in [27]. The formula (11) is a correct version of Theorem 3 [27] for .

## Notes

### Acknowledgements

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.

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