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

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.


Introduction
Throughout this paper, we always make use of the following notation: N denotes the set of natural numbers, N 0 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 a; q 0 1, a; q n n−1 j 0 1 − q j a , n ∈ N, a; q ∞ ∞ j 0 1 − q j a , q < 1, a ∈ C.

1.1
The q-numbers and q-numbers factorial is defined by a q 1 − q a 1 − q q / 1 ; 0 q !1; Discrete Dynamics in Nature and Society respectively.The q-polynomial coefficient is defined by n k q q; q n q; q n−k q; q k .1.3 The q-analogue of the function x y n is defined by In the standard approach to the q-calculus two exponential function are used:

1.5
From this form we easily see that e q z E q −z 1.Moreover, D q e q z e q z , D q E q z E q qz , 1.6 where D q is defined by 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 : t n n! .

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 B α n,q and polynomials B α n,q x, y in x, y of order α are defined by means of the generating function functions: 1.9 Definition 1.2.The q-Genocchi numbers G α n,q and polynomials G α n,q x, y in x, y are defined by means of the generating functions:

1.10
It is obvious that

1.11
Here B α n x and E α n x denote the classical Bernoulli, and Genocchi polynomials of order α are defined by 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 wellknown 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 E α n,q x, y of order α are readily derived from Definition 1.2.We choose to omit the details involved.

1.15
Property 1.6.Differential relations: D q,x G α n,q x, y n q G α n−1,q x, y , D q,y G α n,q x, y n q G α n−1,q x, qy .

Explicit Relationship between the q-Genocchi and the q-Bernoulli Polynomials
In this section we prove an interesting relationship between the q-Genocchi polynomials G α n,q x, y 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 ∈ N 0 , the following relationship holds true between the q-Genocchi and the q-Bernoulli polynomials.
Proof.Using the following identity: 2t e q t 1 α e q tx E q ty 2t e q t 1 α e q tx • e q t/m − 1

2.3
It remains to use Property 1.8.
Since G α n,q x, y 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 ∈ N 0 , the following relationship holds true between the q-Genocchi and the q-Bernoulli polynomials.
Proof.The proof is based on the following identity: 2t e q t 1 α e q tx E q ty 2t e q t 1 α E q ty • e q t/m − 1 t • t e q t/m − 1 • e q t m mx .

2.5
Next we discuss some special cases of Theorems 2.1 and 2.2.By noting that G 0 j,q 0, y q 1/2 j j−1 y j , G 0 j,q x, −1 x − 1 j q , 2.6 we deduce from Theorems 2.1 and 2.2 Corollary 2.3 below.
Corollary 2.3.For n ∈ N 0 , m ∈ N the following relationship holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Corollary 2.4.For n ∈ N 0 , m ∈ N 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 q-Genocchi numbers G α k,q , by setting y 0 in Theorem 2.1, we obtain the following explicit relationship between the q-Genocchi polynomials G α k,q of order α and the q-Bernoulli polynomials.

2.10
Corollary 2.6.For n ∈ N 0 the following relationship holds true: G n,q x, y n k 0 n k q 2 k 1 q k 1 q q 1/2 k k−1 y k − G k 1,q 0, y B n−k,q x, 0 .