1. Preliminaries

Imagine that be a fixed odd prime number. We now start with definition of the following notations. Let be the field -adic rational numbers and let be the completion of algebraic closure of .

Thus,

Then is integral domain, which is defined by or

We assume that with as an indeterminate. The -adic absolute value , is normally defined by where with , and .

is a -extension of , which is defined by we note that (see [1–17]).

Throughout this paper, we use notation of , where denotes set of Natural numbers.

We say that is a uniformly differentiable function at a point , if the difference quotient, has a limit as , and we denote this by . Then, for , we can start with the following expression: which represents a -adic -analogue of Riemann sums for . The integral of on will be defined as the limit of these sums, when it exists. The -adic -integral of function is defined by Kim in [7, 12] as

The bosonic integral is considered as a bosonic limit , . Similarly, the fermionic -adic integral on is introduced by Kim as follows: (for more details, see [13–16]).

From (1.9), it is well-known equality that where (for details, see [2, 3, 8, 9, 12, 13, 15–17]).

The -Genocchi polynomials with wegiht are introduced as

From (1.11), we have where are called -Genocchi numbers with weight . Then, -Genocchi numbers are defined as with the usual convention about replacing by is used (for details, see [3]).

Let be the space of continuous functions on . For , -adic analogue of Bernstein operator for is defined by where . Here, is called -adic Bernstein polynomials, which are defined by (for details, see [1, 4, 5, 7]).

The -Bernoulli polynomials and numbers with weight are defined by Kim et al., respectively, (for more information, see [10]).

The author, by using derivative operator, will investigate some interesting identities on the -Genocchi numbers and polynomials arising from their generating function. Also, the author derives some relations between -Genocchi numbers and -Bernoulli numbers by using Kim’s -Volkenborn integral and fermionic -adic -integral on .

2. Novel Properties of -Genocchi Numbers and Polynomials with Weight 0

Let . Then, by using (1.10), we easily procure the following:

From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s -Genocchi polynomials with weight in [3] as follows:

Here, we assume that is a fixed parameter. Let Thus, by expression of (2.3), we can readily see the following:

Last from equality, taking derivative operator as on the both sides of (2.4), then, we easily see that where and is identity operator. By multiplying on both sides of (2.5), we get

Let us take derivative operator on the both sides of (2.6). Then, we get

Let (not ) be the constant term in a Laurent series of in (2.3). Then, we get

By (2.3), we easily see

We see that the members of (2.11) are proportional to the Bernstein polynomials with the following theorem.

Theorem 2.1. For , one has

Proof. By expressions of (2.8) and (2.9), we see that
By applying basic operations to above equality, we can easily reach to the desired result.

As a special case, we derive the following.

Corollary 2.2. For , one has

Proof. When into (2.10), we derive the following identity: Here, is greatest integer . Then, we complete the proof of Theorem.

From (2.2), we note that

By (2.14) and (1.11), we easily see that

Now, let us consider definition of integral from to in (2.11), then we have where is beta function which is defined by

As a result, we obtain the following theorem.

Theorem 2.3. For , one has

Proof. By taking integral from to in (2.11), we easily reach to desired result.

Substituting into Theorem 2.1, we readily get

By differentiating both sides of (2.11) with respect to , we have the following:

We now give interesting theorem for -Genocchi numbers with weight as follows.

Theorem 2.4. For , one has

Proof. It is proved by using definition of integral on the both sides in the following equality, that is, Last from equality, we discover the following: Then, taking integral from to both sides of last equality, we get
Thus, we complete the proof of the theorem.

Theorem 2.5. For , one has

Proof. In view of (2.2) and (2.23), we discover the following applications:
Thus, we give evidence of the theorem.

As into Theorem 2.5, it leads to the following interesting property.

Corollary 2.6. For , one has where is ordinary Genocchi polynomials, which is defined by the means of the following generating function [9]:

3. Some Identities -Genocchi Numbers and -Bernoulli Numbers by Using Kim’s -Adic -Integrals on

In this section, we consider -Genocchi numbers and -Bernoulli numbers by means of -adic -integral on . Now, we start with the following theorem.

Theorem 3.1. For ,, one has

Proof. For , then by (2.11), On the other hand, the right hand side of (2.11), Combining and , we arrive to the proof of the theorem.

Theorem 3.2. For , one has here .

Proof. Let us take fermionic -adic -inetgral on left-hand side of Theorem 2.5, we get In other word, we consider the right-hand side of Theorem 2.5 as follows:
Equating and , we complete the proof of the theorem.