# On the Values of the Weightedq-Zeta and L-Functions

Authors: T. Kim; S. H. Lee; Hyeon-Ho Han; C. S. Ryoo
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

Recently, the modifiedq-Bernoulli numbers and polynomials are introduced in (D. V. Dolgy et al., in press). These numbers are valuable to study the weighted q-zeta and L-functions.  In this paper,  we study the weighted q-zeta functions and  weighted L-functions from the modified q-Bernoulli numbers and polynomials with weight α.


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## Body

## 1. Introduction

Letq∈ℂ with |q|<1. The modified q-Bernoulli numbers and polynomials with weight α are defined by (1.1)B̃0,q(α)=αq-1log⁡⁡q,(qαB̃q(α)+1)n-B̃n,q(α)={α[α]qifn=1,0ifn>1,
with the usual convention about replacing (B̃q(α))n by B̃n,q(α) (see [1, 2]).Throughout this paper, we use the notation ofq-number as (1.2)[x]q=1-qx1-q,
(see [1–14]).From (1.1), we note that(1.3)B̃n,q(α)=1(1-qα)n∑l=0n(nl)(-1)lαl[αl]q=1(1-q)n[α]qn∑l=0n(nl)(-1)lαl[αl]q.LetF̃q(α)(t)=∑n=0∞B̃n,q(α)tn/n!, Then, by (1.3), we get (1.4)F̃q(α)(t)=αq-1log⁡⁡qe(1/(1-qα))t-αt[α]q∑m=0∞qαme[m]qαt.Let us define the modifiedq-Bernoulli polynomials with weight α as follows:(1.5)B̃n,q(α)(x)=∑l=0n(nl)[x]qαn-lqαlxB̃l,q(α)=([x]qα+qxαB̃q(α))n,
with the usual convention about replacing (B̃q(α))n by B̃n,q(α) (see [1–13]).From (1.5), we can derive the following equation:(1.6)B̃n,q(α)(x)=1(1-qα)n∑l=0n(nl)(-1)lqαlxαl[αl]q=1(1-q)n[α]qn∑l=0n(nl)(-1)lqαlxαl[αl]q,
(see [2]).LetF̃q(α)(t,x)=∑n=0∞B̃n,q(α)(x)tn/n!, then, by (1.6), we get (1.7)F̃q(α)(t,x)=αq-1log⁡⁡qe(1/(1-qα))t-tα[α]q∑m=0∞qα(m+x)e[m+x]qαt.In this paper, we consider the generalizedq-Bernoulli numbers with weight α, and we study the weighted q-zeta function and q-analogue of L-function with weight α from the modified q-Bernoulli numbers and polynomials with weight α.


## 2. Weightedq-Zeta Function and Weighted q-L-Function

From (1.7), we note that(2.1)B̃n,q(α)(x)=α(1-q)n[α]qn(q-1log⁡⁡q)-nα[α]q∑m=0∞qα(m+x)[m+x]qαn-1.
For n∈ℕ, we have(2.2)-B̃n,q(α)(x)n=(α[α]q)(11-qα)n-1(1log⁡⁡q)+α[α]q∑m=0∞qα(m+x)[m+x]qαn-1.LetΓ(s) be the gamma function, then we consider the following complex integral. For s∈ℂ,(2.3)1Γ(s)∫0∞F̃q(α)(-t,x)ts-2dt=αs-1q-1log⁡⁡q(1-qα)s-1+α[α]q∑m=0∞qα(m+x)[m+x]qαs,
where x≠0,-1,-2,-3,….Now, we define the twisted Hurwitz's typeq-zeta function as follows.Fors∈ℂ, define (2.4)ζ̃q(α)(s,x)=α[α]q11-s(1-qα)slog⁡⁡q+α[α]q∑m=0∞qα(m+x)[m+x]qαs,
where x≠0,-1,-2,-3,….Note thatζ̃q(α)(s,x) is meromorphic function whole in complex s-plane except for s=1.From (2.3) and (2.4), we can derive the following equation: (2.5)ζ̃q(α)(s,x)=1Γ(s)∫0∞F̃q(α)(-t,x)ts-2dt.By (1.7), (2.3), (2.4), (2.5), and Laurent series, we get(2.6)ζ̃q(α)(1-k,x)=-B̃k,q(α)(x)k,
where k∈ℕ.Therefore, by (2.6), we obtain the following theorem.Theorem 2.1.
Fork∈ℕ, one has
(2.7)ζ̃q(α)(1-k,x)=-B̃k,q(α)(x)k.
From (2.4), one notes that
(2.8)ζ̃q(α)(s,1)=α[α]q11-s(1-qα)slog⁡⁡q+α[α]q∑m=0∞qα(m+1)[m+1]qαs=α[α]q11-s(1-qα)slog⁡⁡q+α[α]q∑m=1∞qαm[m]qαs.
Now, by (2.8), one defines the weighted q-zeta function as follows:
(2.9)ζ̃q(α)(s)=α[α]q11-s(1-qα)slog⁡⁡q+α[α]q∑m=1∞qαm[m]qαs.=ζ̃q(α)(s,1).
For k∈ℕ, by (1.1) and (1.5), one gets
(2.10)ζ̃q(α)(1-k)=ζ̃q(α)(1-k,1)=-B̃k,q(α)(1)k={-(α[α]q+B̃1,q(α))ifk=1,-B̃k,q(α)kifk>1.
Therefore, by (2.10), one obtains the following corollary.Corollary 2.2.
Fork∈ℕ, one has
(2.11)ζ̃q(α)(1-k)={-(α[α]q+B̃1,q(α))ifk=1,-B̃k,q(α)kifk>1.
Letχ be the Dirichlet's character with conductor d∈ℕ. Let us consider the generalized q-Bernoulli polynomials with weight α as follows:
(2.12)F̃q,χ(α)(t,x)=α[α]qt∑m=0∞χ(m)qα(m+x)e[m+x]qαt=∑n=0∞B̃n,χ,q(α)(x)tnn!.
The sequence B̃n,χ,q(α)(x) will be called the nth generalized q-Bernoulli polynomials with weight α attached to χ.
In the special case,x=0, B̃n,χ,q(α)(0)=B̃n,χ,q(α) are called the nth generalized q-Bernoulli numbers with weight α attached to χ.
From (1.7) and (2.12), one notes that
(2.13)F̃q,χ(α)(t,x)=1[d]q∑a=0d-1χ(a)F̃qd(α)([d]qαt,x+ad).
Thus, by (2.13), one gets
(2.14)B̃n,χ,q(α)(x)=[d]qαn[d]q∑a=0d-1χ(a)B̃n,qd(α)(x+ad).
Therefore, by (2.14), one obtains the following theorem.Theorem 2.3.
Forn∈ℤ+, one has
(2.15)B̃n,χ,q(α)(x)=[d]qαn[d]q∑a=0d-1χ(a)B̃n,qd(α)(x+ad).
In the special case, x=0, one obtains the following corollary.Corollary 2.4.
Forn∈ℤ+, one has
(2.16)B̃n,χ,q(α)=[d]qαn[d]q∑a=0d-1χ(a)B̃n,qd(α)(ad).
Let
(2.17)F̃q,χ(α)(t)=α[α]qt∑m=0∞χ(m)qαme[m]qαt=∑n=0∞B̃n,χ,q(α)tnn!,
then, by (2.12) and (2.17), one easily gets
(2.18)B̃n,χ,q(α)(d)-B̃n,χ,q(α)n=α[α]q∑l=0d-1χ(l)qαl[l]qαn-1.
For s∈ℂ, consider
(2.19)1Γ(s)∫0∞F̃q,χ(α)(-t,x)ts-2dt=α[α]q1Γ(s)∫0∞∑m=0∞χ(m)qα(m+x)e-[m+x]qαtts-1dt=α[α]q∑m=0∞χ(m)qα(m+x)[m+x]qαs1Γ(s)∫0∞e-yys-1dy=α[α]q∑m=0∞χ(m)qα(m+x)[m+x]qαs,
where x≠0,-1,-2,-3,….
Now, one defines Hurwitz's typeq-L-function with weight α as follows. For s∈ℂ,
(2.20)L̃q(α)(s,χ∣x)(-t,x)=α[α]q∑n=0∞χ(n)q(n+x)α[n+x]qαs,
where x≠0,-1,-2,-3,….
From (2.19) and (2.20), one notes that
(2.21)L̃q(α)(s,χ∣x)=1Γ(s)∫0∞F̃q,χ(α)(-t,x)ts-2dt.
By (1.7) and (2.21) and Laurent series, one obtains the following theorem.Theorem 2.5.
Fork∈ℕ, one has
(2.22)L̃q(α)(1-k,χ∣x)=-B̃k,χ,q(α)(x)k.
In the special case,x=0, L̃q(α)(1-k,χ∣0)=L̃q(α)(1-k,χ) are called the q-L-function with weight α.
Let(2.23)F̃q(α)(s,a∣F)=α[F]q[α]q(∑m≡a(modF)m>0∞qαm[m]qαs+(1-qα)sF(1-s)log⁡⁡q)=α[F]q[α]q(∑n=0∞qα(a+nF)[a+nF]qαs+(1-qα)sF(1-s)log⁡⁡q)=[F]qα[F]q[F]qαsζ̃qF(α)(s,aF),
where a and F are positive integers with 0<a<F.
Then, by (2.23), one gets
(2.24)H̃q(α)(1-n,a∣F)=-[F]qαnB̃n,χ,q(α)(a/F)[F]qn,n≥1,
and H̃q(α)(s,a∣F) has as simple pole as s=1 with residue (α/[F]q)((q-1)/logqF).
Letχ be the Dirichlet character with conductor F, then one easily sees that
(2.25)L̃q(α)(s,χ)=∑a=1Fχ(a)H̃q(α)(s,a∣F).



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*Source: 476381-2011-11-17.xml*