Ozden Advances in Difference Equations 2013, 2013:40
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RESEARCH Open Access
q-Dirichlet type L-functions with weight α
Hacer Ozden*
*Correspondence:
hozden@uludag.edu.tr Abstract
Department of Mathematics,
Faculty of Art and Science, The aim of this paper is to construct q-Dirichlet type L-functions with weight α. We
University of Uludag, Bursa, Turkey 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
Keywords: generalized Bernoulli polynomials; Dirichlet L-function; Hurwitz zeta
function; generalized q-Bernoulli numbers with weight α
1 Introduction
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: N = {, , . . .}, N = {, , , . . .} =
N ∪ {}, +Z = {, , , . . .}, –Z = {–,–, . . .}. 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 ln(z) denotes the principal branch of the multi-valued function ln(z) with the imagi-
nary part (ln(z)) constrained by –π < (ln(z))≤ π .
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:N = {, , . . .},N =
{, , , . . .} =N∪ {}, Z+ = {, , , . . .}, Z– = {–,–, . . .}. 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 f ∈N. The Dirichlet L-function
is defined as follows:
∑∞ χ (n)
L(s,χ ) =
ns
, ()
n=
where s ∈ C ((s) > ) (see [–] and the references cited in each of earlier works). The
function L(s,χ ) is analytically continued to the complex s-plane, one has
B
L( – n, ) = – n,χχ , ()
n
© 2013 Ozden; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-
tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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where n ∈ +Z andBn,χ denotes the usual generalized Bernoulli numbers, which are defined
by means of the following generating function (see [–]):
∑f – χ (a)eatt ∑∞ tn
ft = Bn,χ .
a= e –  = n!n
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 gen-
eralized Bernoulli polynomials with weight α as follows:
t ∑∞ ∑∞ n
F (α)
α
(x, t,χ ) = qα(x+m)χ (m)et[x+m]
t
qα
q = B̃
(α)
n,χ ,q(x) , ()[α]q = = n!m n
where
∈ (∣∣ ∣q qαC ∣ )<  .
Remark . By substituting χ ≡  into (), we have
∑∞ ∑∞ n
F (α)
αt
(x, t) = qα(x+m)et[x+m]qα = B̃(α)
t
q n,χ ,q(x) ,[α]q = = n!m n
which is defined by Kim [].
Remark . By substituting α =  into (), we have
lim B̃(α)→ n,χ ,q(x) = Bn,χ (x),q 
where Bn,χ (x) denotes generalized Bernoulli polynomials attached to Drichlet character χ
with conductor f ∈N (see [–]).
By applying the derivative operator
∣
∂k ∣
F (α)(x, t)∣
∂tk q ∣t=
to (), we obtain
kα ∑∞
qα(x+m)χ (m)[m + x]k– (α)
[α] qα
= B̃k,χ ,q(x), ()
q m=
where
∣∣ ∣qα∣ < .
Observe that when χ ≡  in (), one can obtain recurrence relation for the polynomial
B̃(α)k,q(x).
By using (), we define a two-variable q-Dirichlet L-function with weight α as follows.
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Definition . Let s,q ∈ C (|qα| < ). The two-variable q-Dirichlet L-functions with
weight α are defined by
(α) | –α
∑∞ qα(x+m)χ (m)
L̃q (s,χ x) = . ()[α] sq [m + x]m= qα
Remark . Substituting x =  into (), then the q-Dirichlet L-functions with weight α
are defined by
∑∞ α(m+)
L̃(α)
–α q χ (m)
q (s,χ |) = .[α]q = ( + qα[m])sm
Remark . By applying the Mellin transformation to (), Kim et al. [] defined two-
variable q-Dirichlet L-functions with weight α as follows: Let |q| <  and (s) > , then
 ∫ ∞ ( { } )
L̃(α)(s,χ |x) = ts–q F (α)q (x, –t)dt min (s),(x) >  .(s) 
For x = , by using (), we obtain the following corollary.
Corollary . Let q, s ∈C.We assume that (q) <  and |qα| < . Then we have
( )
(α) | –α( – q
α)s ∑∞ ∑∞ n + s – 
L̃q (s,χ ) = χ (m)q
αn(m+).
[α]q m= n= n
Remark . Substituting α =  into () and then q→ , we have
∑∞
| χ (m)L̃(s,χ x) = –
= (m + x)
s
m
= –L(s,χ |x),
which gives us a two-variable Dirichlet L-function (see [, , , –, ]). Substituting
x =  into the above equation, one has ().
Theorem . Let k ∈ +Z . Then we have
B̃(α) (x)
L̃(α)( – k, |x) = – k,χ ,qq χ . ()k
Proof By substituting s =  – k with k ∈ +Z into (), we have
∑∞
L̃(α)q ( – k,χ |
–α
x) = qα(x+m)χ (m)[m + x]k–.
[α] qαq m=
Combining () with the above equation, we arrive at the desired result. 
Remark . If q→ , then () reduces to ().
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Remark . Substituting χ =  into (), we modify Kim’s et al. zeta function as follows
(see []):
–α ∑∞ qα[m+x]
–ζ̃ (α) (
( )
α)
q (s,x) = L̃q (s, |x) = [α] [m + x]s (s) >  . ()q m= qα
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 . Let n ∈ +Z . Then we have
B̃(α)(x)
ζ̃ (α)q ( – n,x) = –
n,q . ()
n
Nowwe are ready to give relationship between () and (). Substitutingm = a+kn, where
a = , , . . . ,k; n = , , , . . . into (), we obtain
–α ∑k ∑∞ qα(x+a+kn)χ (a + kn)
L̃(α)q (s,χ |x) = [α]q = = [a + kn + x]sa n qα
– ∑k ∑∞α knα
= qα(x+a)
q
χ (a)
[α] sq = = [k]qα [
a+x
k + n]
s
a n ( q
αk
)
–α [α] kqαk ∑= α(x+a) (kα) a + x
[α]q[k]s αk
q χ (a)ζ̃
α qkα s, .
qα a= k
Therefore, we have the following theorem.
Theorem . The following relation holds true:
–α–kα[α] αk ∑k ( )
L̃(α)
a + x
q (s, |x) = q qα(x+a) (a) (kα)χ s χ ζ̃qkα s, . ()[α]q[k]qα a= k
By substituting s = –n with n ∈ +Z into () and combining with () and (), we give ex-
plicitly a formula of the generalized Bernoulli polynomials with weight α by the following
theorem.
Theorem . The following formula holds true:
–kα ( )
( ) α [α] ∑kαkB̃ α (x) = q α(x+a) (α) a + xn,χ ,q –n q χ (a)B̃n,q . ()[α]q[k]qα a= k
By using (), we obtain the following corollary.
Corollary . The following formula holds true:
( )
α–kα[α] [k]n– ∑k ∑n ( )(α) αk α n a + x n–jB̃n,χ ,q(x) = q q qα(x+a)χ (a) B̃(α).[α] k j,qq a= j= j
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Competing interests
The author declares that she has no competing interests.
Authors’ contributions
The author completed the paper herself. The author read and approved the final manuscript.
Acknowledgements
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.
Received: 5 December 2012 Accepted: 5 February 2013 Published: 21 February 2013
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Cite this article as: Ozden: q-Dirichlet type L-functions with weight α. Advances in Difference Equations 2013 2013:40.