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On a class of q-Bernoulli and q-Euler polynomials
Advances in Difference Equations volume 2013, Article number: 108 (2013)
The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava-Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.
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-shifted factorial is defined by
The q-numbers and q-numbers 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 .
Over 70 years ago, Carlitz extended the classical Bernoulli and Euler numbers and polynomials and introduced the q-Bernoulli and the q-Euler numbers and polynomials (see [2, 3] and ). There are numerous recent investigations on this subject by, among many other authors, Cenkci 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. , Srivastava , Mahmudov .
We first give here the definitions of the q-Bernoulli and the q-Euler polynomials of higher order as follows.
Definition 1 Let , . The q-Bernoulli numbers and polynomials in x, y of order α are defined by means of the generating functions:
Definition 2 Let , . The q-Euler numbers and polynomials in x, y of order α are defined by means of the generating functions:
It is obvious that
Here and denote the classical Bernoulli and Euler polynomials of order α which are defined by
In fact, Definitions 1 and 2 define two different types and of the q-Bernoulli polynomials and two different types and of the q-Euler polynomials. Both polynomials and ( and ) coincide with the classical higher-order Bernoulli polynomials (Euler polynomials) in the limiting case .
For the q-Bernoulli numbers and the q-Euler numbers of order n, we have
respectively. Note that the q-Bernoulli numbers are defined and studied in .
The aim of the present paper is to obtain some results for the above newly defined q-Bernoulli and q-Euler polynomials. It should be mentioned that q-Bernoulli and q-Euler polynomials in our definitions are polynomials of x and y, and when , they are polynomials of x, but in other definitions they are functions of . First advantage of this approach is that for , () becomes the classical Bernoulli (Euler ) polynomial, and we may obtain the q-analogues of well-known results, for example, those of Srivastava and Pintér , Cheon , etc. The second advantage is that we find the relation between q-Bernstein polynomials and Phillips q-Bernoulli polynomials and derive the formulas involving the q-Stirling numbers of the second kind, q-Bernoulli polynomials and Phillips q-Bernstein polynomials.
2 Preliminaries and lemmas
In this section we provide some basic formulas for the q-Bernoulli and q-Euler polynomials in order 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 (Addition theorems)
For all , we have
In particular, setting and in (1) and (2), we get the following formulas for q-Bernoulli and q-Euler polynomials, respectively
Setting and in (1) and (2), we get
Clearly, (5) and (6) are q-analogues of
Lemma 4 We have
Lemma 5 (Difference equations)
From (7) and (5), (8) and (6), we obtain the following formulas.
Lemma 6 We have
Putting in (9) and (10) and noting that
we arrive at the following expansions:
which are q-analogues of the following familiar expansions:
Lemma 7 (Recurrence relationships)
The polynomials and satisfy the following difference relationships:
3 Explicit relationship between the q-Bernoulli and q-Euler polynomials
In this section we investigate some explicit relationships between the q-Bernoulli and q-Euler polynomials. Here some q-analogues of known results are given. We also obtain new formulas and some of their special cases below. These formulas are some extensions of the formulas of Srivastava and Pintér, Cheon and others.
Theorem 8 For , the following relationships hold true:
Using the following identity:
It is clear that
On the other hand,
It remains to use the formula (12). □
Next we discuss some special cases of Theorem 8.
Corollary 9 For , , the relationship
holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Corollary 10 ()
For , , the following relationship holds true:
Corollary 11 For , the following relationship holds true:
Corollary 12 For , the following relationship holds true:
Theorem 13 For , the relationships
hold true between the q-Bernoulli polynomials and q-Euler polynomials.
The proof is based on the following identities:
and is similar to that of Theorem 8. □
Next we discuss some special cases of Theorem 13.
Corollary 14 For , , the relationship
holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Corollary 15 ()
For , , the following relationships hold true:
Corollary 16 For , the following relationship holds true:
Corollary 17 For , the following relationships hold true:
These formulas are q-analogues of the formula of Srivastava and Pintér .
4 q-Stirling numbers and q-Bernoulli polynomials
In this section, we aim to derive several formulas involving the q-Bernoulli polynomials, the q-Euler polynomials of order α, the q-Stirling numbers of the second kind and the q-Bernstein polynomials.
Theorem 18 Each of the following relationships holds true for the Stirling numbers of the second kind:
The familiar q-Stirling numbers of the second kind are defined by
where . Next we give the relationship between q-Bernstein basis defined by Phillips  and q-Bernoulli polynomials
Theorem 19 We have
The proof follows from the following identities:
Finally, in their limit case when , this last result (19) would reduce to the following formula for the classical Bernoulli polynomials and the Bernstein basis :
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Dedicated to Professor Hari M Srivastava.
The author sincerely thanks the anonymous referees for their valuable suggestions and comments which have helped improve this article.
The author declares that he has no competing interests.