Discrete Dynamics in Nature and Society
Volume 2012, Issue 1 169348
Research Article
Open Access

q-Analogues of the Bernoulli and Genocchi Polynomials and the Srivastava-Pintér Addition Theorems

N. I. Mahmudov, 

Corresponding Author

N. I. Mahmudov

Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey emu.edu.tr

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First published: 06 September 2012
https://doi.org/10.1155/2012/169348
Citations: 7
Academic Editor: Lee Chae Jang

Abstract

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli and Genocchi 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.

1. Introduction

Throughout this paper, we always make use of the following notation: ℕ denotes the set of natural numbers, ℕ0 denotes the set of nonnegative integers, ℝ denotes the set of real numbers, and ℂ denotes the set of complex numbers.

The q-shifted factorial is defined by
(1.1)
The q-numbers and q-numbers factorial is defined by
(1.2)
respectively. The q-polynomial coefficient is defined by
(1.3)
The q-analogue of the function (x+y)n is defined by
(1.4)
In the standard approach to the q-calculus two exponential function are used:
(1.5)
From this form we easily see that eq(z)Eq(−z) = 1. Moreover,
(1.6)
where Dq is defined by
(1.7)
The previous q-standard notation can be found in [1].
Carlitz has introduced the q-Bernoulli numbers and polynomials in [2]. Srivastava and Pintér proved some relations and theorems between the Bernoulli polynomials and Euler polynomials in [3]. They also gave some generalizations of these polynomials. In [4–6], Kim et al. investigated some properties of the q-Euler polynomials and Genocchi polynomials. They gave some recurrence relations. In [7], Cenkci et al. gave the q-extension of Genocchi numbers in a different manner. In [5], Kim gave a new concept for the q-Genocchi numbers and polynomials. In [8], Simsek et al. investigated the q-Genocchi zeta function and l-function by using generating functions and Mellin transformation. We also recall the definitions of the q-Bernoulli and the q-Genocchi polynomials of higher order (see [2, 9–12]):
(1.8)
We propose the following definitions. We define the q-Bernoulli and the q-Genocchi polynomials of higher order in two variables x and y, using two q-exponential functions, which helps us easily prove some properties of these polynomials and q-analogue of the Srivastava and Pintér addition theorem.

Definition 1.1. The q-Bernoulli numbers and polynomials in x, y of order α are defined by means of the generating function functions:

(1.9)

Definition 1.2. The q-Genocchi numbers and polynomials in x, y are defined by means of the generating functions:

(1.10)

It is obvious that
(1.11)
Here and denote the classical Bernoulli, and Genocchi polynomials of order α are defined by
(1.12)

The aim of the present paper is to obtain some results for the q-Genocchi polynomials (properties of the q-Bernoulli polynomials are studied in [13]). The q-analogues of well-known results, for example, Srivastava and Pintér [3], can be derived from these q-identities. It should be mentioned that probabilistic proofs the Srivastava-Pintér addition theorems were given recently in [14]. The formulas involving the q-Stirling numbers of the second kind, q-Bernoulli polynomials and q-Bernstein polynomials, are also given. Furthermore some special cases are also considered.

The following elementary properties of the q-Genocchi polynomials of order α are readily derived from Definition 1.2. We choose to omit the details involved.

Property 1.3. Special values of the q-Genocchi polynomials of order α:

(1.13)

Property 1.4. Summation formulas for the q-Genocchi polynomials of order α:

(1.14)

Property 1.5. Difference equations:

(1.15)

Property 1.6. Differential relations:

(1.16)

Property 1.7. Addition theorem of the argument:

(1.17)

Property 1.8. Recurrence relationships:

(1.18)

2. Explicit Relationship between the q-Genocchi and the q-Bernoulli Polynomials

In this section we prove an interesting relationship between the q-Genocchi polynomials of order α and the q-Bernoulli polynomials. Here some q-analogues of known results will be given. We also obtain new formulas and their some special cases in the following.

Theorem 2.1. For n ∈ ℕ0, the following relationship

(2.1)
holds true between the q-Genocchi and the q-Bernoulli polynomials.

Proof. Using the following identity:

(2.2)
we have
(2.3)
It remains to use Property 1.8.

Since is not symmetric with respect to x and y, we can prove a different form of the previously mentioned theorem. It should be stressed out that Theorems 2.1 and 2.2 coincide in the limiting case when q → 1−.

Theorem 2.2. For n ∈ ℕ0, the following relationship

(2.4)
holds true between the q-Genocchi and the q-Bernoulli polynomials.

Proof. The proof is based on the following identity:

(2.5)

Next we discuss some special cases of Theorems 2.1 and 2.2. By noting that
(2.6)
we deduce from Theorems 2.1 and 2.2 Corollary 2.3 below.

Corollary 2.3. For n ∈ ℕ0, m ∈ ℕ the following relationship

(2.7)
holds true between the q-Bernoulli polynomials and q-Euler polynomials.

Corollary 2.4. For n ∈ ℕ0, m ∈ ℕ the following relationship holds true:

(2.8)
(2.9)
between the classical Genocchi polynomials and the classical Bernoulli polynomials.

Note that the formula (2.9) is new for the classical polynomials.

In terms of the q-Genocchi numbers , by setting y = 0 in Theorem 2.1, we obtain the following explicit relationship between the q-Genocchi polynomials of order α and the q-Bernoulli polynomials.

Corollary 2.5. The following relationship holds true:

(2.10)

Corollary 2.6. For n ∈ ℕ0 the following relationship holds true:

(2.11)

Corollary 2.7. For n ∈ ℕ0 the following relationship holds true:

(2.12)

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