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q-Dirichlet type L-functions with weight α
Advances in Difference Equations volume 2013, Article number: 40 (2013)
The aim of this paper is to construct q-Dirichlet type L-functions with weight α. We give the values of these functions at negative integers. These values are related to the generalized q-Bernoulli numbers with weight α.
AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.
Recently Kim, Simsek, Yang and also many mathematicians have studied a two-variable Dirichlet L-function.
In this paper, we need the following standard notions: , , , . Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers. We assume that denotes the principal branch of the multi-valued function with the imaginary part constrained by .
In this paper, we study the two-variable Dirichlet L-function with weight α. We give some properties of this function. We also give explicit values of this function at negative integers which are related to the generalized Bernoulli polynomials and numbers with weight α.
Throughout this presentation, we use the following standard notions: , , , . Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers.
Let χ be a primitive Dirichlet character with conductor . The Dirichlet L-function is defined as follows:
2 Two-variable q-Dirichlet L-function with weight α
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
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
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:
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Dedicated to Professor Hari M Srivastava.
The present investigation was supported by the Commission of Scientific Research Projects of Uludag University, project number UAP(F) 2011/38 and UAP(F) 2012-16. We would like to thank the referees for their valuable comments.
The author declares that she has no competing interests.
The author completed the paper herself. The author read and approved the final manuscript.
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Ozden, H. q-Dirichlet type L-functions with weight α. Adv Differ Equ 2013, 40 (2013). https://doi.org/10.1186/1687-1847-2013-40
- generalized Bernoulli polynomials
- Dirichlet L-function
- Hurwitz zeta function
- generalized q-Bernoulli numbers with weight α