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: