On the Values of the Weighted 𝑞-Zeta and 𝐿-Functions

Abstract

Recently, the modified 𝑞-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.

1. Introduction

Let 𝑞∈ℂ with |𝑞|<1. The modified 𝑞-Bernoulli numbers and polynomials with weight 𝛼 are defined by 𝐵^(𝛼)_0,𝑞=𝛼𝑞−1,𝑞log𝑞^𝛼𝐵_𝑞^(𝛼)+1^𝑛−𝐵^(𝛼)_𝑛,𝑞=𝛼[𝛼]_𝑞if0𝑛=1,if𝑛>1, (1.1) with the usual convention about replacing (𝐵_𝑞^(𝛼))^𝑛 by 𝐵^(𝛼)_𝑛,𝑞 (see [1, 2]).

Throughout this paper, we use the notation of 𝑞-number as [𝑥]_𝑞=1−𝑞^𝑥,1−𝑞 (1.2) (see [1–14]).

From (1.1), we note that𝐵^(𝛼)_𝑛,𝑞=11−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝛼𝑙[𝛼𝑙]_𝑞=11−𝑞^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝛼𝑙[𝛼𝑙]_𝑞. (1.3)

Let 𝐹_𝑞^(𝛼)(𝑡)=∑^∞_𝑛=0𝐵^(𝛼)_𝑛,𝑞𝑡^𝑛/𝑛!, Then, by (1.3), we get 𝐹_𝑞^(𝛼)(𝑡)=𝛼𝑞−1𝑒log𝑞^{(1/(1−𝑞𝛼))𝑡}−𝛼𝑡[𝛼]_𝑞^∞_𝑚=0𝑞^𝛼𝑚𝑒^{[𝑚]𝑞𝛼𝑡}. (1.4)

Let us define the modified 𝑞-Bernoulli polynomials with weight 𝛼 as follows:𝐵^(𝛼)_𝑛,𝑞(𝑥)=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑥𝐵^(𝛼)_𝑙,𝑞=[𝑥]_𝑞𝛼+𝑞^𝑥𝛼𝐵_𝑞^(𝛼)𝑛, (1.5) with the usual convention about replacing (𝐵_𝑞^(𝛼))^𝑛 by 𝐵^(𝛼)_𝑛,𝑞 (see [1–13]).

From (1.5), we can derive the following equation:𝐵^(𝛼)_𝑛,𝑞(𝑥)=11−𝑞^𝛼𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥𝛼𝑙[𝛼𝑙]_𝑞=11−𝑞^𝑛[𝛼]^𝑛_𝑞^𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝛼𝑙𝑥𝛼𝑙[𝛼𝑙]_𝑞, (1.6) (see [2]).

Let 𝐹_𝑞^(𝛼)(𝑡,𝑥)=∑^∞_𝑛=0𝐵^(𝛼)_𝑛,𝑞(𝑥)𝑡^𝑛/𝑛!, then, by (1.6), we get 𝐹_𝑞^(𝛼)(𝑡,𝑥)=𝛼𝑞−1𝑒log𝑞^{(1/(1−𝑞𝛼))𝑡}𝛼−𝑡[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}𝑒^{[𝑚+𝑥]𝑞𝛼𝑡}. (1.7)

In this paper, we consider the generalized 𝑞-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. Weighted 𝑞-Zeta Function and Weighted 𝑞-𝐿-Function

From (1.7), we note that𝐵^(𝛼)_𝑛,𝑞(𝑥)=𝛼1−𝑞^𝑛[𝛼]^𝑛_𝑞𝑞−1−log𝑞𝑛𝛼[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}[𝑚+𝑥]_𝑞^𝑛−1_𝛼. (2.1) For 𝑛∈ℕ, we have−𝐵^(𝛼)_𝑛,𝑞(𝑥)𝑛=𝛼[𝛼]_𝑞11−𝑞^𝛼𝑛−11+𝛼log𝑞[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}[𝑚+𝑥]_𝑞^𝑛−1_𝛼. (2.2)

Let Γ(𝑠) be the gamma function, then we consider the following complex integral. For 𝑠∈ℂ,1Γ(𝑠)^∞₀𝐹_𝑞^(𝛼)(−𝑡,𝑥)𝑡^𝑠−2𝛼𝑑𝑡=𝑠−1𝑞−1log𝑞1−𝑞^𝛼𝑠−1+𝛼[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}[𝑚+𝑥]^𝑠_𝑞𝛼, (2.3) where 𝑥≠0,−1,−2,−3,….

Now, we define the twisted Hurwitz's type 𝑞-zeta function as follows.

For 𝑠∈ℂ, define 𝜁̃_𝑞^(𝛼)(𝑠,𝑥)=𝛼[𝛼]_𝑞11−𝑠1−𝑞^𝛼𝑠+𝛼log𝑞[𝛼]_𝑞^∞_𝑚=0𝑞^{𝛼(𝑚+𝑥)}[𝑚+𝑥]^𝑠_𝑞𝛼, (2.4) where 𝑥≠0,−1,−2,−3,….

Note that 𝜁̃_𝑞^(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: 𝜁̃_𝑞^(𝛼)(𝑠,𝑥)=1Γ(𝑠)^∞₀𝐹_𝑞^(𝛼)(−𝑡,𝑥)𝑡^𝑠−2𝑑𝑡. (2.5)

By (1.7), (2.3), (2.4), (2.5), and Laurent series, we get𝜁̃_𝑞^(𝛼)()𝐵1−𝑘,𝑥=−^{(𝛼)𝑘,𝑞}(𝑥)𝑘, (2.6) where 𝑘∈ℕ.

Therefore, by (2.6), we obtain the following theorem.

Theorem 2.1. For 𝑘∈ℕ, one has 𝜁̃_𝑞^(𝛼)()𝐵1−𝑘,𝑥=−^{(𝛼)𝑘,𝑞}(𝑥)𝑘. (2.7) From (2.4), one notes that 𝜁̃_𝑞^(𝛼)(𝑠,1)=𝛼[𝛼]_𝑞11−𝑠1−𝑞^𝛼𝑠+𝛼log𝑞[𝛼]_𝑞^∞_𝑚=0𝑞^𝛼(𝑚+1)[𝑚+1]^𝑠_𝑞𝛼=𝛼[𝛼]_𝑞11−𝑠1−𝑞^𝛼𝑠+𝛼log𝑞[𝛼]_𝑞^∞_𝑚=1𝑞^𝛼𝑚[𝑚]^𝑠_𝑞𝛼. (2.8) Now, by (2.8), one defines the weighted 𝑞-zeta function as follows: 𝜁̃_𝑞^(𝛼)(𝑠)=𝛼[𝛼]_𝑞11−𝑠1−𝑞^𝛼𝑠+𝛼log𝑞[𝛼]_𝑞^∞_𝑚=1𝑞^𝛼𝑚[𝑚]^𝑠_𝑞𝛼.=𝜁̃_𝑞^(𝛼)(𝑠,1). (2.9) For 𝑘∈ℕ, by (1.1) and (1.5), one gets 𝜁̃_𝑞^(𝛼)(1−𝑘)=𝜁̃_𝑞^(𝛼)()𝐵1−𝑘,1=−^{(𝛼)𝑘,𝑞}(1)𝑘=−𝛼[𝛼]_𝑞+𝐵^(𝛼)_1,𝑞if−𝐵𝑘=1,_{(𝛼)𝑘,𝑞}𝑘if𝑘>1. (2.10) Therefore, by (2.10), one obtains the following corollary.

Corollary 2.2. For 𝑘∈ℕ, one has 𝜁̃_𝑞^(𝛼)()=−𝛼1−𝑘[𝛼]_𝑞+𝐵^(𝛼)_1,𝑞if−𝐵𝑘=1,_{(𝛼)𝑘,𝑞}𝑘if𝑘>1. (2.11)
Let 𝜒 be the Dirichlet's character with conductor 𝑑∈ℕ. Let us consider the generalized 𝑞-Bernoulli polynomials with weight 𝛼 as follows: 𝐹^(𝛼)_𝑞,𝜒()=𝛼𝑡,𝑥[𝛼]_𝑞𝑡^∞_𝑚=0𝜒(𝑚)𝑞^{𝛼(𝑚+𝑥)}𝑒^{[𝑚+𝑥]𝑞𝛼𝑡}=^∞_𝑛=0𝐵^(𝛼)_{𝑛,𝜒,𝑞}(𝑥)𝑡^𝑛.𝑛! (2.12) The sequence 𝐵^(𝛼)_{𝑛,𝜒,𝑞}(𝑥) will be called the 𝑛th generalized 𝑞-Bernoulli polynomials with weight 𝛼 attached to 𝜒.
In the special case, 𝑥=0, 𝐵^(𝛼)_{𝑛,𝜒,𝑞}(0)=𝐵^(𝛼)_{𝑛,𝜒,𝑞} are called the 𝑛th generalized 𝑞-Bernoulli numbers with weight 𝛼 attached to 𝜒.
From (1.7) and (2.12), one notes that 𝐹^(𝛼)_𝑞,𝜒()=1𝑡,𝑥[𝑑]_𝑞^𝑑−1_𝑎=0𝜒(𝑎)𝐹_𝑞^(𝛼)_𝑑[𝑑]_𝑞𝛼𝑡,𝑥+𝑎𝑑. (2.13) Thus, by (2.13), one gets 𝐵^(𝛼)_{𝑛,𝜒,𝑞}(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝜒(𝑎)𝐵^(𝛼)_{𝑛,𝑞𝑑}𝑥+𝑎𝑑. (2.14) Therefore, by (2.14), one obtains the following theorem.

Theorem 2.3. For 𝑛∈ℤ₊, one has 𝐵^(𝛼)_{𝑛,𝜒,𝑞}(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝜒(𝑎)𝐵^(𝛼)_{𝑛,𝑞𝑑}𝑥+𝑎𝑑. (2.15) In the special case, 𝑥=0, one obtains the following corollary.

Corollary 2.4. For 𝑛∈ℤ₊, one has 𝐵^(𝛼)_{𝑛,𝜒,𝑞}=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝜒(𝑎)𝐵^(𝛼)_{𝑛,𝑞𝑑}𝑎𝑑. (2.16) Let 𝐹^(𝛼)_𝑞,𝜒(𝑡)=𝛼[𝛼]_𝑞𝑡^∞_𝑚=0𝜒(𝑚)𝑞^𝛼𝑚𝑒^{[𝑚]𝑞𝛼𝑡}=^∞_𝑛=0𝐵^(𝛼)_{𝑛,𝜒,𝑞}𝑡^𝑛,𝑛! (2.17) then, by (2.12) and (2.17), one easily gets 𝐵^(𝛼)_{𝑛,𝜒,𝑞}(𝑑)−𝐵^(𝛼)_{𝑛,𝜒,𝑞}𝑛=𝛼[𝛼]_𝑞^𝑑−1_𝑙=0𝜒(𝑙)𝑞^𝛼𝑙[𝑙]_𝑞^𝑛−1_𝛼. (2.18) For 𝑠∈ℂ, consider 1Γ(𝑠)^∞₀𝐹^(𝛼)_𝑞,𝜒(−𝑡,𝑥)𝑡^𝑠−2𝛼𝑑𝑡=[𝛼]_𝑞1Γ(𝑠)^∞₀^∞_𝑚=0𝜒(𝑚)𝑞^{𝛼(𝑚+𝑥)}𝑒^{−[𝑚+𝑥]𝑞𝛼𝑡}𝑡^𝑠−1=𝛼𝑑𝑡[𝛼]_𝑞^∞_𝑚=0𝜒(𝑚)𝑞^{𝛼(𝑚+𝑥)}[𝑚+𝑥]^𝑠_𝑞𝛼1Γ(𝑠)^∞₀𝑒^−𝑦𝑦^𝑠−1=𝛼𝑑𝑦[𝛼]_𝑞^∞_𝑚=0𝜒(𝑚)𝑞^{𝛼(𝑚+𝑥)}[𝑚+𝑥]^𝑠_𝑞𝛼, (2.19) where 𝑥≠0,−1,−2,−3,….
Now, one defines Hurwitz's type 𝑞-𝐿-function with weight 𝛼 as follows. For 𝑠∈ℂ, 𝐿_𝑞^(𝛼)()=𝛼𝑠,𝜒∣𝑥−𝑡,𝑥[𝛼]_𝑞^∞_𝑛=0𝜒(𝑛)𝑞^{(𝑛+𝑥)𝛼}[𝑛+𝑥]^𝑠_𝑞𝛼, (2.20) where 𝑥≠0,−1,−2,−3,….
From (2.19) and (2.20), one notes that 𝐿_𝑞^(𝛼)=1𝑠,𝜒∣𝑥Γ(𝑠)^∞₀𝐹^(𝛼)_𝑞,𝜒(−𝑡,𝑥)𝑡^𝑠−2𝑑𝑡. (2.21)
By (1.7) and (2.21) and Laurent series, one obtains the following theorem.

Theorem 2.5. For 𝑘∈ℕ, one has 𝐿_𝑞^(𝛼)𝐵1−𝑘,𝜒∣𝑥=−^{(𝛼)𝑘,𝜒,𝑞}(𝑥)𝑘. (2.22)
In the special case, 𝑥=0, 𝐿_𝑞^(𝛼)(1−𝑘,𝜒∣0)=𝐿_𝑞^(𝛼)(1−𝑘,𝜒) are called the 𝑞-𝐿-function with weight 𝛼.
Let 𝐹_𝑞^(𝛼)()=𝛼𝑠,𝑎∣𝐹[𝐹]_𝑞[𝛼]_𝑞^∞_{𝑚≡𝑎(mod𝐹)𝑚>0}𝑞^𝛼𝑚[𝑚]^𝑠_𝑞𝛼+1−𝑞^𝛼𝑠𝐹()=𝛼1−𝑠log𝑞[𝐹]_𝑞[𝛼]_𝑞^∞_𝑛=0𝑞^{𝛼(𝑎+𝑛𝐹)}[𝑎+𝑛𝐹]^𝑠_𝑞𝛼+1−𝑞^𝛼𝑠𝐹()=[𝐹]1−𝑠log𝑞_𝑞𝛼[𝐹]_𝑞[𝐹]^𝑠_𝑞𝛼𝜁̃_𝑞^(𝛼)_𝐹𝑠,𝑎𝐹, (2.23) where 𝑎 and 𝐹 are positive integers with 0<𝑎<𝐹.
Then, by (2.23), one gets 𝐻_𝑞^(𝛼)()[𝐹]1−𝑛,𝑎∣𝐹=−^𝑛𝑞𝛼𝐵^(𝛼)_{𝑛,𝜒,𝑞}(𝑎/𝐹)[𝐹]_𝑞𝑛,𝑛≥1, (2.24) and 𝐻_𝑞^(𝛼)(𝑠,𝑎∣𝐹) has as simple pole as 𝑠=1 with residue (𝛼/[𝐹]_𝑞)((𝑞−1)/log𝑞^𝐹).
Let 𝜒 be the Dirichlet character with conductor 𝐹, then one easily sees that 𝐿_𝑞^(𝛼)𝑠,𝜒=^𝐹_𝑎=1𝜒(𝑎)𝐻_𝑞^(𝛼)(𝑠,𝑎∣𝐹). (2.25)