q-Dirichlet type L-functions with weight α
© Ozden; licensee Springer 2013
Received: 5 December 2012
Accepted: 5 February 2013
Published: 21 February 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.
2 Two-variable q-Dirichlet L-function with weight α
which is defined by Kim .
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.
For , by using (5), we obtain the following corollary.
Combining (4) with the above equation, we arrive at the desired result. □
Remark 2.9 If , then (6) reduces to (1).
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.
Therefore, we have the following theorem.
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.
By using (10), we obtain the following corollary.
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.
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