Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 169348, 8 pages

doi:10.1155/2012/169348

Research Article

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

N. I. Mahmudov

Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey

Received 24 April 2012; Revised 5 July 2012; Accepted 23 July 2012

Academic Editor: Lee Chae Jang

Copyright © 2012 N. I. Mahmudov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli and Genocchi polynomials based on the -integers. The -analogues of well-known formulas are derived. The -analogue of the Srivastava-Pintér addition theorem is obtained.

1. Introduction

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

The -shifted factorial is defined by

The -numbers and -numbers factorial is defined by

respectively. The -polynomial coefficient is defined by

The -analogue of the function is defined by

In the standard approach to the -calculus two exponential function are used:

From this form we easily see that . Moreover,

where is defined by

The previous -standard notation can be found in [1].

Carlitz has introduced the -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 -Euler polynomials and Genocchi polynomials. They gave some recurrence relations. In [7], Cenkci et al. gave the -extension of Genocchi numbers in a different manner. In [5], Kim gave a new concept for the -Genocchi numbers and polynomials. In [8], Simsek et al. investigated the -Genocchi zeta function and -function by using generating functions and Mellin transformation. We also recall the definitions of the -Bernoulli and the -Genocchi polynomials of higher order (see [2, 9–12]):

We propose the following definitions. We define the -Bernoulli and the -Genocchi polynomials of higher order in two variables and , using two -exponential functions, which helps us easily prove some properties of these polynomials and -analogue of the Srivastava and Pintér addition theorem.

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

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

It is obvious that

Here and denote the classical Bernoulli, and Genocchi polynomials of order are defined by

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

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

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

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

Property 1.5. Difference equations:

Property 1.6. Differential relations:

Property 1.7. Addition theorem of the argument:

Property 1.8. Recurrence relationships:

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

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

Theorem 2.1. For , the following relationship

holds true between the -Genocchi and the -Bernoulli polynomials.

Proof. Using the following identity:

we have

It remains to use Property 1.8.

Since is not symmetric with respect to and , 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 .

Theorem 2.2. For , the following relationship

holds true between the -Genocchi and the -Bernoulli polynomials.

Proof. The proof is based on the following identity:

Next we discuss some special cases of Theorems 2.1 and 2.2. By noting that

we deduce from Theorems 2.1 and 2.2 Corollary 2.3 below.

Corollary 2.3. For , the following relationship

holds true between the -Bernoulli polynomials and -Euler polynomials.

Corollary 2.4. For , the following relationship holds true:

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 -Genocchi numbers , by setting in Theorem 2.1, we obtain the following explicit relationship between the -Genocchi polynomials of order and the -Bernoulli polynomials.

Corollary 2.5. The following relationship holds true:

Corollary 2.6. For the following relationship holds true:

Corollary 2.7. For the following relationship holds true:

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