In this section, we give some new relations and identities related to -Genocchi numbers and polynomials. Firstly we give some generating functions of the -Genocchi numbers, which were defined by Kim [3, 10, 11]:

and let

(cf.[3, 10, 11, 23]), where denotes -Genocchi numbers.

We note that -Genocchi numbers, were defined by Kim [3, 10, 11].

By using the above generating functions, -Genocchi polynomials, , are defined by means of the following generating function:

Our generating function of is similar to that of [3, 12, 21, 23]. By using Cauchy product in (2.3), we easily obtain

Then by comparing coefficients of on both sides of the above equation, for we obtain the following result.

Theorem 2.1.

Let be an integer with Then one has

By using the same method in [3, 12, 21] in (2.3), we have

and after some elementary calculations, we have

By comparing coefficients of on both sides of the above equation, we arrive at the following corollary.

Corollary 2.2.

Let . Then one has

We give some of -Genocchi polynomials as follows:

From the generating function we have the following.

Corollary 2.3.

Let . Then one has

Proof of the Corollary 2.3 was given by Kim [3, 12]. We give some of -Genocchi numbers as follows: ,

Observe that if then

By using derivative operator to (2.6), we have

After some elementary calculations, we arrive at the following corollary.

Corollary 2.4.

Let be a positive integer. Then one has

Corollary 2.5.

Let be a positive integer. Then one has

Proof.

Proof of this corollary is easily obtained from (2.4).

Generalized -Genocchi numbers are defined by means of the following generating function (this generating function is similar to that of [3, 12, 21–24]):

where denotes the Dirichlet character with conductor the set of positive integers.

Observe that when (2.13) reduces to (2.3).

By (2.13), we have

After some elementary calculations and by comparing coefficients on both sides of the above equation, we get

By setting , where and , in the above equation, we obtain

In [15], Srivastava et al. defined the following generalized Barnes-type Changhee -Bernoulli numbers.

Let be the Dirichlet character with conductor . Then the generalized Barnes-type Changhee -Bernoulli numbers with attached to are defined as follows:

(cf. [15]). Substituting and into the above equation, we have

By using derivative operator to the above, we obtain

By substituting (2.9) and (2.19) into (2.16), after some calculations, we arrive at the following theorem.

Theorem 2.6.

Let be the Dirichlet character with conductor . If is odd, then one has

if is even, then one has

where is defined in (2.19).

Remark 2.7.

In Theorem 2.6, we give new relations between generalized -Genocchi numbers, with attached to , -Genocchi numbers, , and Barnes-type Changhee -Bernoulli numbers. For detailed information about generalized Barnes-type Changhee -Bernoulli numbers with attached to see [15].

Generalized Genocchi polynomials are defined by means of the following generating function:

Theorem 2.8.

Let be the Dirichlet character with conductor . Then one has

Remark 2.9.

Generating functions of and are different from those of [3, 12, 22, 23]. Kim defined generating function of as follows [12]:

In [21], Simsek defined generating function of by