1. Introduction, Definitions, and Notations

Let be the complex number field. We assume that with and that the -number is defined by in this paper.

Many mathematicians have studied -Bernoulli, -Euler polynomials, and related topics (see [1–23]). It is known that the Bernoulli polynomials are defined by

(1.1)

and that are called the th Bernoulli numbers.

The recurrence formula for the classical Bernoulli numbers is as follows,

(1.2)

(see [1, 3, 23]). The -extension of the following recurrence formula for the Bernoulli numbers is

(1.3)

with the usual convention of replacing by (see [5, 7, 14]).

Now, by introducing the following well-known identities

(1.4)

(see [6]).

The generating functions of the second kind Stirling numbers and -Bernstein polynomials, respectively, can be defined as follows,

(1.5)
(1.6)

(see [2]), where (see [4]).

Throughout this paper, , , , , and will respectively denote the ring of rational integers, the field of rational numbers, the ring -adic rational integers, the field of -adic rational numbers, and the completion of the algebraic closure of . Let be the normalized exponential valuation of such that . If , we normally assume or so that for (see [7–19]).

In this study, we present a new generating function related to the -Bernoulli numbers and -Bernoulli polynomials and give a new construction of these numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. We also consider the generalized -Bernoulli polynomials attached to Dirichlet's character and have their generating function. We obtain distribution relations for the -Bernoulli polynomials and have some identities involving -Bernoulli numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of this generating function which interpolates the -Bernoulli polynomials at negative integers and are associated with -Bernstein polynomials.

2. New Approach to -Bernoulli Numbers and Polynomials

Let be the set of natural numbers and . For with , let us define the -Bernoulli polynomials as follows,

(2.1)

Note that

(2.2)

where are classical Bernoulli polynomials. In the special case , are called the th -Bernoulli numbers. That is,

(2.3)

From (2.1) and (2.3), we note that

(2.4)

From (2.1) and (2.3), we can easily derive the following equation:

(2.5)

Equations (2.4) and (2.5), we see that and

(2.6)

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has

(2.7)

with the usual convention of replacing and .

From (2.1), one notes that

(2.8)

Therefore, one obtains the following theorem.

Theorem 2.2.

For , one has

(2.9)

By (2.1), one sees that

(2.10)

By (2.1) and (2.10), one obtains the following theorem.

Theorem 2.3.

For , one has

(2.11)

From (2.11) one can derive that, for ,

(2.12)

By (2.12), one sees that, for ,

(2.13)

Therefore, one obtains the following theorem.

Theorem 2.4.

For , one has

(2.14)

In (2.9), substitute instead of , one obtains

(2.15)

which is the relation between -Bernoulli polynomials, -Bernoulli numbers, and -Bernstein polynomials. In (1.5), substitute instead of , one gets

(2.16)

In (2.16), substitute instead of , and putting the result in (2.15), one has the following theorem.

Theorem 2.5.

For and , one has

(2.17)

where and are the second kind Stirling numbers and -Bernstein polynomials, respectively.

Let be Dirichlet's character with . Then, one defines the generalized -Bernoulli polynomials attached to as follows,

(2.18)

In the special case , are called the th generalized -Bernoulli numbers attached to . Thus, the generating function of the generalized -Bernoulli numbers attached to are as follows,

(2.19)

By (2.1) and (2.18), one sees that

(2.20)

Therefore, one obtains the following theorem.

Theorem 2.6.

For and , one has

(2.21)

By (2.18) and (2.19), one sees that

(2.22)

Hence,

(2.23)

For , one now considers the Mellin transformation for the generating function of . That is,

(2.24)

for , and .

From (2.24), one defines the zeta type function as follows,

(2.25)

Note that is an analytic function in the whole complex -plane. Using the Laurent series and the Cauchy residue theorem, one has

(2.26)

By the same method, one can also obtain the following equations:

(2.27)

For ,one defines Dirichlet type --function as

(2.28)

where . Note that is also a holomorphic function in the whole complex -plane. From the Laurent series and the Cauchy residue theorem, one can also derive the following equation:

(2.29)

In (2.23), substitute instead of , one obtains

(2.30)

which is the relation between the th generalized -Bernoulli numbers and -Bernoulli polynomials attached to and -Bernstein polynomials. From (2.16), one has the following theorem.

Theorem 2.7.

For and , one has

(2.31)

One now defines particular -zeta function as follows,

(2.32)

From (2.32), one has

(2.33)

where is given by (2.25). By (2.26), one has

(2.34)

Therefore, one obtains the following theorem.

Theorem 2.8.

For , we have

(2.35)