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

Note that

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

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

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

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

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has

with the usual convention of replacing and .

From (2.1), one notes that

Therefore, one obtains the following theorem.

Theorem 2.2.

For , one has

By (2.1), one sees that

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

Theorem 2.3.

For , one has

From (2.11) one can derive that, for ,

By (2.12), one sees that, for ,

Therefore, one obtains the following theorem.

Theorem 2.4.

For , one has

In (2.9), substitute instead of , one obtains

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

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

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,

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,

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

Therefore, one obtains the following theorem.

Theorem 2.6.

For and , one has

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

Hence,

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

for , and .

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

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

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

For ,one defines Dirichlet type --function as

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:

In (2.23), substitute instead of , one obtains

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

One now defines particular -zeta function as follows,

From (2.32), one has

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

Therefore, one obtains the following theorem.

Theorem 2.8.

For , we have