The following generating functions are given by Kim et al.  and are related to the generalized Bernoulli polynomials with weight α as follows:
Remark 2.1 By substituting into (3), we have
which is defined by Kim .
Remark 2.2 By substituting into (3), we have
where denotes generalized Bernoulli polynomials attached to Drichlet character χ with conductor (see [1–22]).
By applying the derivative operator
to (3), we obtain
Observe that when in (4), one can obtain recurrence relation for the polynomial .
By using (4), we define a two-variable q-Dirichlet L-function with weight α as follows.
Definition 2.3 Let (). The two-variable q-Dirichlet L-functions with weight α are defined by
Remark 2.4 Substituting into (5), then the q-Dirichlet L-functions with weight α are defined by
Remark 2.5 By applying the Mellin transformation to (3), Kim et al.  defined two-variable q-Dirichlet L-functions with weight α as follows: Let and , then
For , by using (5), we obtain the following corollary.
Corollary 2.6 Let . We assume that and . Then we have
Remark 2.7 Substituting into (5) and then , we have
which gives us a two-variable Dirichlet L-function (see [6, 11, 16, 18–20, 22]). Substituting into the above equation, one has (2).
Theorem 2.8 Let . Then we have
Proof By substituting with into (5), we have
Combining (4) with the above equation, we arrive at the desired result. □
Remark 2.9 If , then (6) reduces to (1).
Remark 2.10 Substituting into (5), we modify Kim’s et al. zeta function as follows (see ):
This function gives us Hurwitz-type zeta functions with weight α. It is well known that this function interpolates the q-Bernoulli polynomials with weight α at negative integers, which is given by the following lemma.
Lemma 2.11 Let . Then we have
Now we are ready to give relationship between (7) and (5). Substituting , where ; into (5), we obtain
Therefore, we have the following theorem.
Theorem 2.12 The following relation holds true:
By substituting with into (9) and combining with (8) and (6), we give explicitly a formula of the generalized Bernoulli polynomials with weight α by the following theorem.
Theorem 2.13 The following formula holds true:
By using (10), we obtain the following corollary.
Corollary 2.14 The following formula holds true: