On a class of generalized q-Bernoulli and q-Euler polynomials
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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.
KeywordsNonnegative Integer Correct Form Euler Number Basic Formula Addition Theorem
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 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, 3, 4]. There are numerous recent investigations on this subject by, among many other authors, Cenki et al. [5, 6, 7], Choi et al.  and , Kim et al. [10, 11, 12, 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.
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.
It is a correct form of formula (2.7) from  for .
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.
hold true between the generalized q-Bernoulli polynomials and q-Euler polynomials.
Next we discuss some special cases of Theorem 11.
holds true between the generalized q-Bernoulli polynomials and the q-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 .
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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