A note on Carlitz q-Bernoulli numbers and polynomials

Abstract

In this article, we first aim to give simple proofs of known formulae for the generalized Carlitz q-Bernoulli polynomials β_m,χ(x, q) in the p-adic case by means of a method provided by Kim and then to derive a complex, analytic, two-variable q-L-function that is a q-analog of the two-variable L-function. Using this function, we calculate the values of two-variable q-L-functions at nonpositive integers and study their properties when q tends to 1.

Mathematics Subject Classification (2000): 11B68; 11S80.

1. Introduction

Let p be a fixed prime. We denote by ℤ _p , ℚ _p , and ℂ _p the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of ℚ _p , respectively. Let v _p be the normalized exponential valuation of ℂ _p with ${\left|p\right|}_p=p^{-v_p\left(p\right)}=p^{-1}$. When one talks of a q-extension, q can be variously considered as an indeterminate, a complex number q ∈ ℂ, or a p-adic number q ∈ ℂ _p . If q ∈ ℂ, one normally assumes |q| < 1. If q ∈ ℂ _p , one normally assumes |1 - q| _p < p^-1/(p-1), so that q^x= exp(x log _p q) for |x| _p ≤ 1.

Let d be a fixed positive integer. Let

$$\begin{array}{ll}\hfill X=X_d& =\underset{\stackrel{⃖}{N}}{\text{lim}}\left(\mathbb{Z}/d{p}^N\mathbb{Z}\right),X_1={\mathbb{Z}}_p,\\ \hfill X^{*}& =\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup }a+dp{\mathbb{Z}}_p,\\ \hfill a+d{p}^N{\mathbb{Z}}_p& =\left\{x\in X|x\equiv a\left(modd{p}^N\right)\right\},\end{array}$$

(1.1)

where a ∈ ℤ lies in 0 ≤ a < dp^N. We use the following notation:

$${\left[x\right]}_q=\frac{1-q^x}{1-q}.$$

(1.2)

Hence lim_q→1 [x] _q = x for any x ∈ ℂ in the complex case and any x with |x| _p ≤ 1 in the present p-adic case. This is the hallmark of a q-analog: The limit as q → 1 recovers the classical object.

In 1937, Vandiver [1] and, in 1941, Carlitz [2] discussed generalized Bernoulli and Euler numbers. Since that time, many authors have studied these and other related subjects (see, e.g., [3–6]). The final breakthrough came in the 1948 article by Carlitz [7]. He defined inductively new q-Bernoulli numbers β _m = β _m (q) by

$${\beta }_0\left(q\right)=1,q{\left(q\beta \left(q\right)+1\right)}^m-{\beta }_m\left(q\right)=\left\{\begin{array}{cc}1\hfill & \mathsf{\text{if}}m=1\hfill \\ 0\hfill & \mathsf{\text{if}}m>1,\hfill \end{array}\right.$$

(1.3)

with the usual convention of βⁱby β _i . The q-Bernoulli polynomials are defined by

$${\beta }_m\left(x,q\right)={\left(q^x\beta \left(q\right)+{\left[x\right]}_q\right)}^m=\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\beta }_i\left(q\right)q^{ix}{\left[x\right]}_q^{m-i}.$$

(1.4)

In 1954, Carlitz [8] generalized a result of Frobenius [3] and showed many of the properties of the q-Bernoulli numbers β _m (q). In 1964, Carlitz [9] extended the Bernoulli, Eulerian, and Euler numbers and corresponding polynomials as a formal Dirichlet series. In what follows, we shall call them the Carlitz q-Bernoulli numbers and polynomials.

Some properties of Carlitz q-Bernoulli numbers β _m (q) were investigated by various authors. In [10], Koblitz constructed a q-analog of p-adic L-functions and suggested two questions. Question (1) was solved by Satoh [11]. He constructed a complex analytic q-L-series that is a q-analog of Dirichlet L-function and interpolates Carlitz q-Bernoulli numbers, which is an answer to Koblitz's question. By using a q-analog of the p-adic Haar distribution (see (1.6) below), Kim [12] answered part of Koblitz's question (2) and constructed q-analogs of the p-adic log gamma functions G_p,q(x) on ℂ _p \ ℤ _p .

In [11], Satoh constructed the generating function of the Carlitz q-Bernoulli numbers F _q (t) in ℂ which is given by

$$F_q\left(t\right)=\sum _{m=0}^{\infty }{q}^m{e}^{{\left[m\right]}_{q}t}\left(1-q-q^{m}t\right)=\sum _{m=0}^{\infty }{\beta }_m\left(q\right)\frac{t^m}{m!},$$

(1.5)

where q is a complex number with 0 < |q| < 1. He could not explicitly determine F _q (t) in ℂ _p , see [11, p.347].

In [12], Kim defined the q-analog of the p-adic Haar distribution μ_Haar(a + p^Nℤ _p ) = 1/p^Nby

$${\mu }_q\left(a+p^N{\mathbb{Z}}_p\right)=\frac{q^a}{{\left[p^N\right]}_q}.$$

(1.6)

Using this distribution, he proved that the Carlitz q-Bernoulli numbers β _m (q) can be represented as the p-adic q-integral on ℤ _p by μ _q , that is,

$${\beta }_m\left(q\right)={\int }_{{\mathbb{Z}}_p}{\left[a\right]}_q^{m}d{\mu }_q\left(a\right),$$

(1.7)

and found the following explicit formula

$${\beta }_m\left(q\right)=\frac{1}{{\left(q-1\right)}^m}\sum _{i=0}^m{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right)\frac{i+1}{{\left[i+1\right]}_q},$$

(1.8)

where m ≥ 0 and q ∈ ℂ _p with $0<{\left|1-q\right|}_p<p^{-\frac{1}{p-1}}$.

Recently, Kim and Rim [13] constructed the generating function of the Carlitz q-Bernoulli numbers F _q (t) in ℂ _p :

$$F_q\left(t\right)=e^{\frac{t}{1-q}}\sum _{j=0}^{\infty }\frac{j+1}{{\left[j+1\right]}_q}{\left(-1\right)}^j{\left(\frac{1}{1-q}\right)}^j\frac{t^j}{j!},$$

(1.9)

where q ∈ ℂ _p with $0<{\left|1-q\right|}_p<p^{-\frac{1}{p-1}}$.

This article is organized as follows.

In Section 2, we consider the generalized Carlitz q-Bernoulli polynomials in the p-adic case by means of a method provided by Kim. We obtain the generating functions of the generalized Carlitz q-Bernoulli polynomials. We shall provide some basic formulas for the generalized Carlitz q-Bernoulli polynomials which will be used to prove the main results of this article.

In Section 3, we construct the complex, analytic, two-variable q-L-function that is a q- analog of the two-variable L-function. Using this function, we calculate the values of two-variable q-L-functions at nonpositive integers and study their properties when q tends to 1.

2. Generalized Carlitz q-Bernoulli polynomials in the p-adic (and complex) case

For any uniformly differentiable function f : ℤ _p → ℂ _p , the p-adic q-integral on ℤ _p is defined to be the limit $\frac{1}{{\left[p^N\right]}_q}{\sum }_{a=0}^{p^N-1}f\left(a\right)q^a$ as N → ∞. The uniform differentiability guarantees the limit exists. Kim [12, 14–16] introduced this construction, denoted I _q (f), where |1 - q| _p < p^-1/(p-1).

The construction of I _q (f) makes sense for many q in ℂ _p with the weaker condition |1 - q| _p < 1. Indeed, when |1 - q| _p < 1 the function q^xis uniformly differentiable and the space of uniformly differentiable functions ℤ _p → ℂ _p is closed under multiplication, so we can make sense of its p-adic q-integral I _q (f) for |1 - q| _p < 1.

Lemma 2.1. For q ∈ ℂ _p with 0 < |1 - q| _p < 1 and x ∈ ℤ _p , we have

$$\underset{N\to \infty }{\text{lim}}\frac{1}{1-q^{p^N}}\sum _{a=0}^{p^N-1}{q}^{ax}=\frac{x}{1-q^x}.$$

Proof. We assume that q ∈ ℂ _p satisfies the condition 0 < |1 - q| _p < 1. Then it is known that

$$q^x=\sum _{m=0}^{\infty }\left(\begin{array}{c}x\hfill \\ m\hfill \end{array}\right){\left(q-1\right)}^m$$

for any x ∈ ℤ _p (see [[17], Lemma 3.1 (iii)]). Therefore, we obtain

$$\begin{array}{ll}\hfill \underset{N\to \infty }{\text{lim}}\frac{1}{1-q^{p^N}}\sum _{a=0}^{p^N-1}{q}^{ax}& =\frac{1}{1-q^x}\underset{N\to \infty }{\text{lim}}\frac{{\left(q^{p^N}\right)}^x-1}{q^{p^N}-1}\\ =\frac{1}{1-q^x}\underset{N\to \infty }{\text{lim}}\frac{{\sum }_{m=1}^{\infty }\left(\begin{array}{c}x\hfill \\ m\hfill \end{array}\right){\left(q^{p^N}-1\right)}^m}{q^{p^N}-1}\\ =\frac{1}{1-q^x}\underset{N\to \infty }{\text{lim}}\sum _{m=0}^{\infty }\left(\begin{array}{c}x\hfill \\ m+1\hfill \end{array}\right){\left(q^{p^N}-1\right)}^m\\ =\frac{x}{1-q^x}.\end{array}$$

This completes the proof.

Definition 2.2 ([12, §2, p. 323]). Let χ be a primitive Dirichlet character with conductor d ∈ ℕ and let x ∈ ℤ _p . For q ∈ ℂ _p with 0 < |1 - q| _p < 1 and an integer m ≥ 0, the generalized Carlitz q-Bernoulli polynomials β_m,χ(x, q) are defined by

$$\begin{array}{ll}\hfill {\beta }_{m,\chi }\left(x,q\right)& ={\int }_X\chi \left(a\right){\left[x+a\right]}_q^{m}d{\mu }_q\left(a\right)\\ =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[d{p}^N\right]}_q}\sum _{a=0}^{d{p}^N-1}\chi \left(a\right){\left[x+a\right]}_q^m{q}^a.\end{array}$$

(2.1)

Remark 2.3. If χ = χ⁰, the trivial character and x = 0, then (2.1) reduces to (1.7) since d = 1. In particular, Kim [12] defined a class of p-adic interpolation functions G _p,q (x) of the Carlitz q-Bernoulli numbers β _m (q) and gave several interesting applications of these functions.

By Lemma 2.1, we can prove the following explicit formula of β_m,χ(x, q) in ℂ _p .

Proposition 2.4. For q ∈ ℂ _p with 0 < |1 - q| _p < 1 and an integer m ≥ 0, we have

$${\beta }_{m,\chi }\left(x,q\right)=\frac{1}{{\left(1-q\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_q}.$$

Proof. For m ≥ 0, (2.1) implies

$$\begin{array}{ll}\hfill {\beta }_{m,\chi }\left(x,q\right)& =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[d\right]}_q}\frac{1}{{\left[p^N\right]}_{q^d}}\sum _{k=0}^{d-1}\sum _{a=0}^{p^N-1}\chi \left(k+da\right){\left[x+k+da\right]}_q^m{q}^{k+da}\\ =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left(1-q\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right)\frac{q^k}{1-q^{d{p}^N}}\sum _{q=0}^{p^N-1}{\left(1-q^{x+k+da}\right)}^m{q}^{da}\\ =\frac{1}{{\left(1-q\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\\ \times \underset{N\to \infty }{\text{lim}}\frac{1}{1-{\left(q^d\right)}^{p^N}}\sum _{a=0}^{p^N-1}{\left(q^d\right)}^{a\left(i+1\right)}\\ =\frac{1}{{\left(1-q\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{i+1}{1-q^{d\left(i+1\right)}}\\ \mathsf{\text{(where}}\mathsf{\text{we}}\mathsf{\text{use}}\mathsf{\text{Lemma}}\mathsf{\text{2}}\mathsf{\text{.1).}}\end{array}$$

This completes the proof.

Remark 2.5. We note here that similar expressions to those of Proposition 2.4 with χ = χ⁰ are given by Kamano [[18], Proposition 2.6] and Kim [12, §2]. Also, Ryoo et al. [19, Theorem 4] gave the explicit formula of β_m,χ(0, q) in ℂ for m ≥ 0.

Lemma 2.6. Let χ be a primitive Dirichlet character with conductor d ∈ ℕ. Then for q ∈ ℂ with |q| < 1,

$$\sum _{m=0}^{\infty }\chi \left(m\right)q^{mx}=\frac{1}{1-q^{dx}}\sum _{k=0}^{d-1}\chi \left(k\right)q^{kx}.$$

Proof. If we write m = ad + k, where 0 ≤ k ≤ d - 1 and a = 0,1, 2,..., we have the desired result.

We now consider the case:

$$q\in \overline{\mathbb{Q}}\cap {ℂ}_p,0<\left|q\right|<1,0<{\left|1-q\right|}_p<1.$$

(2.2)

For instance, if we set

$$q=\frac{1}{1-pz}\in \overline{\mathbb{Q}}\cap {ℂ}_p$$

for each z ≠ 0 ∈ ℤ and p > 3, we find 0 < |q| < 1, 0 < |1 - q| _p < 1.

Let F_q,χ(t, x) be the generating function of β_m,χ(x, q) defined in Definition 2.2. From Proposition 2.4, we have

$$\begin{array}{ll}\hfill F_{q,\chi }\left(t,x\right)& =\sum _{m=0}^{\infty }{\beta }_{m,\chi }\left(x,q\right)\frac{t^m}{m!}\\ =\sum _{m=0}^{\infty }\left(\frac{1}{{\left(1-q\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_q}\right)\frac{t^m}{m!}\\ =P_{q,\chi }\left(t,x\right)+Q_{q,\chi }\left(t,x\right),\end{array}$$

(2.3)

where

$$P_{q,\chi }\left(t,x\right)=\sum _{m=0}^{\infty }\frac{1}{{\left(1-q\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{i}{{\left[d\left(i+1\right)\right]}_q}\frac{t^m}{m!}$$

and

$$Q_{q,\chi }\left(t,x\right)=\sum _{m=0}^{\infty }\frac{1}{{\left(1-q\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{1}{{\left[d\left(i+1\right)\right]}_q}\frac{t^m}{m!}.$$

Then, noting that

$$e^{\frac{t}{1-q}}=\sum _{i=0}^{\infty }{\left(-1\right)}^i{\left(q-1\right)}^{-i}\frac{t^i}{i!},$$

we see that

$$\begin{array}{ll}\hfill P_{q,\chi }\left(t,x\right)& =\sum _{m=0}^{\infty }\frac{1}{{\left(1-q\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{i}{{\left[d\left(i+1\right)\right]}_q}\frac{t^m}{m!}\\ =\sum _{n=0}^{\infty }\frac{1}{{\left(1-q\right)}^n}\frac{t^n}{n!}\sum _{j=0}^{\infty }\frac{1}{{\left(q-1\right)}^j}\sum _{k=0}^{d-1}\chi \left(k\right)q^{j\left(x+k\right)+k}\frac{j}{{\left[d\left(j+1\right)\right]}_q}\frac{t^j}{j!}\\ =e^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^j\sum _{k=0}^{d-1}\chi \left(k\right)q^{j\left(x+k\right)+k}\frac{j}{{\left[d\left(j+1\right)\right]}_q}\frac{t^j}{j!}.\end{array}$$

(2.4)

Moreover, (2.4) now becomes

$$\begin{array}{ll}\hfill P_{q,\chi }\left(t,x\right)& =e^{\frac{t}{1-q}}\sum _{j=1}^{\infty }{\left(\frac{1}{q-1}\right)}^j\sum _{k=0}^{d-1}\chi \left(k\right)q^{j\left(x+k\right)+k}\frac{1}{{\left[d\left(j+1\right)\right]}_q}\frac{t^j}{\left(j-1\right)!}\\ =e^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^j{q}^{\left(j+1\right)x}\sum _{k=0}^{d-1}\chi \left(k\right)\frac{q^{k\left(j+2\right)}}{{q^d}^{\left(j+2\right)}-1}\frac{t^{j+1}}{j!}\\ =-t{e}^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^j{q}^{\left(j+1\right)x}\sum _{n=0}^{\infty }\chi \left(n\right)q^{n\left(j+2\right)}\frac{t^j}{j!}\\ \mathsf{\text{(where}}\mathsf{\text{we}}\mathsf{\text{use}}\mathsf{\text{Lemma}}\mathsf{\text{2}}\mathsf{\text{.6)}}\\ =-t{e}^{\frac{t}{1-q}}\sum _{n=0}^{\infty }\chi \left(n\right)q^{x+2n}\sum _{j=0}^{\infty }{\left(\frac{-q^{n+x}}{1-q}\right)}^j\frac{t^j}{j!}\\ =-t{e}^{\frac{t}{1-q}}\sum _{n=0}^{\infty }\chi \left(x\right)q^{x+2n}{e}^{\frac{\left(-{q^n}^{+x}\right)t}{1-q}}\\ =-t\sum _{n=0}^{\infty }\chi \left(x\right)q^{x+2n}{e}^{{\left[n+x\right]}_{q}t}\end{array}$$

(2.5)

(cf. [13, 16, 20]). Similar arguments apply to the case Q_q,χ(t, x). We can rewrite

$$Q_{q,\chi }\left(t,x\right)=e^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^j\sum _{k=0}^{d-1}\chi \left(k\right)q^{j\left(x+k\right)+k}\frac{1}{{\left[d\left(j+1\right)\right]}_q}\frac{t^j}{j!}$$

(2.6)

and

$$Q_{q,\chi }\left(t,x\right)=\left(1-q\right)\sum _{n=0}^{\infty }\chi \left(n\right)q^n{e}^{{\left[n+x\right]}_{q}t}.$$

(2.7)

Then, by (2.4), (2.5), (2.6), and (2.7), we have the following theorem.

Theorem 2.7. Let$q\in \overline{\mathbb{Q}}\cap {ℂ}_p,0<\left|q\right|<1,0<{\left|1-q\right|}_p<1$. Then the generalized Carlitz q-Bernoulli polynomials β_m,χ(x, q) for m ≤ 0 is given by equating the coefficients of powers of t in the following generating function:

$$\begin{array}{ll}\hfill F_{q,\chi }\left(t,x\right)& =e^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^{j-1}\sum _{k=0}^{d-1}\chi \left(k\right)q^{j\left(x+k\right)+k}\frac{j+1}{q^{d\left(j+1\right)}-1}\frac{t^j}{j!}\\ =\sum _{n=0}^{\infty }\chi \left(n\right)q^n{e}^{{\left[n+x\right]}_{q}t}\left(1-q-q^{n+x}t\right).\end{array}$$

(2.8)

Remark 2.8. If χ = χ 0, the trivial character, and x = 0, (2.8) reduces to (1.5).

3. q-analog of the two-variable L-function (in ℂ)

From Theorem 2.7, for k ≥ 0, we obtain the following

$$\begin{array}{ll}\hfill {\beta }_{k,\chi }\left(x,q\right)& ={\left.{\left(\frac{\mathsf{\text{d}}}{\mathsf{\text{d}}t}\right)}^k{F}_{q,\chi }\left(t,x\right)\right|}_{t=0}\\ =\left(1-q\right)\sum _{m=0}^{\infty }\chi \left(m\right)q^m{\left[m+x\right]}_q^k-k\sum _{m=0}^{\infty }\chi \left(m\right)q^{x+2m}{\left[m+x\right]}_q^{k-1}.\end{array}$$

(3.1)

Hence we can define a q-analog of the L-function as follows:

Definition 3.1. Suppose that χ is a primitive Dirichlet character with conductor d ∈ ℕ. Let q be a complex number with 0 < |q| < 1, and let L _q (s, x, χ) be a function of two-variable (s, x) ∈ ℂ × ℝ defined by

$$L_q\left(s,x,\chi \right)=\frac{1-q}{s-1}\sum _{m=0}^{\infty }\frac{\chi \left(m\right)q^m}{{\left[m+x\right]}_q^{s-1}}+\sum _{m=0}^{\infty }\frac{\chi \left(m\right)q^{m+2x}}{{\left[m+x\right]}_q^s}$$

(3.2)

for 0 < x ≤ 1 (cf. [11, 13, 14, 21–25]).

In particular, the two-variable function L _q (s, x, χ) is a generalization of the one-variable L _q (s, χ) of Satoh [11], yielding the one-variable function when the second variable vanishes.

Proposition 3.2. For k ∈ ℤ, k ≥ 1, the limiting value lim_{s → k}L _q (1 - s, x, χ) = L _q (1 - k, x, χ) exists and is given explicitly by

$$L_q\left(1-k,x,\chi \right)=-\frac{1}{k}{\beta }_{k,\chi }\left(x,q\right).$$

Proof. The proof is clear by Proposition 2.4, Theorem 2.7 and (3.1).

The formula of Proposition 3.2 is slight extension of the result in [19] and [11, Theorem 2].

Theorem 3.3. For any positive integer k, we have

$$\begin{array}{ll}\hfill \underset{q\to 1}{\text{lim}}{\beta }_{k,\chi }\left(x,q\right)& =\underset{q\to 1}{\text{lim}}\frac{1}{{\left(1-q\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^k\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^i{q}^{i\left(x+k\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_q}\\ =B_{k,\chi }\left(x\right),\end{array}$$

where the B_k,χ(x) are the kth generalized Bernoulli polynomials.

Proof. We follow the proof in [[26], Theorem 1] motivated by the study of a simple q-analog of the Riemann zeta function. Recall that the ordinary Bernoulli polynomials B _k (x) are defined by

$$\frac{i}{q^i-1}{q}^{ix}=\frac{1}{\text{log}q}\frac{i\text{log}q}{{e^i}^{\text{log}q}-1}{e}^{x\left(i\text{log}q\right)}=\frac{1}{\text{log}q}\sum _{k=0}^{\infty }{B}_k\left(x\right)i^k\frac{{\left(\text{log}q\right)}^k}{k!},$$

(3.3)

where it is noted that in this instance, the notation B _k (x) is used to replace B^k(x) symbolically. For each m ≥ 1, let

$${\left(e^t-1\right)}^m=\sum _{k=0}^{\infty }{d}_k^{\left(m\right)}\frac{t^k}{k!}.$$

(3.4)

Note that

$${\left(e^t-1\right)}^m=\sum _{i=0}^m{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right)e^{it}=\sum _{k=0}^{\infty }\left(\sum _{i=0}^m\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^{m-i}{i}^k\right)\frac{t^k}{k!}.$$

(3.5)

From (3.4) and (3.5), we obtain

$$d_k^{\left(m\right)}=\left\{\begin{array}{cc}{\sum }_{i=0}^m{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right)i^k,\hfill & m\le k\hfill \\ 0,\hfill & 0\le k<m.\hfill \end{array}\right.$$

(3.6)

It is also clear from the definition that $d_0^{\left(0\right)}=1,d_k^{\left(0\right)}=0$ and $d_k^{\left(k\right)}=k!$ for k ∈ ℕ. From (2.3), (3.3), and (3.6), we obtain

$$\begin{array}{ll}\hfill {\beta }_{m,\chi }\left(x,q\right)& =\frac{q^{-x}}{{\left(q-1\right)}^m}\sum _{k=0}^{d-1}\chi \left(k\right)q^{k+x}\sum _{i=0}^m{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right)q^{i\left(k+x\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_q}\\ =\frac{q^{-x}}{{\left(q-1\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right)\sum _{i=0}^m{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right)\\ \times {e^{d\left(i+1\right)}}^{\text{log}{q}^{\frac{\left(k+x\right)}{d}}}\frac{d\left(i+1\right)\text{log}q}{e^{d\left(i+1\right)\text{log}q-1}}\frac{1}{d\text{log}q}\\ =\frac{q^{-x}}{{\left(q-1\right)}^{m-1}}\sum _{n=0}^{\infty }\left(\sum _{i=0}^m{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(i+1\right)}^n\right)\\ \times d^{n-1}\sum _{k=0}^{d-1}\chi \left(k\right)B_n\left(\frac{k+x}{d}\right)\frac{{\left(\text{log}q\right)}^{n-1}}{n!}\\ =q^{-x}\frac{{\left(\text{log}q\right)}^{m-1}}{{\left(q-1\right)}^{m-1}}{d}^{m-1}\sum _{k=0}^{d-1}\chi \left(k\right)B_m\left(\frac{k+x}{d}\right)\\ +q^{-x}\sum _{\sigma =1}^{\infty }\sum _{i=0}^{\sigma }\left(\begin{array}{c}m+\sigma \hfill \\ i\hfill \end{array}\right)d_{m+\sigma -i}^{\left(m\right)}\frac{1}{\left(m+\sigma \right)!}\frac{{\left(\text{log}q\right)}^{m+\sigma -1}}{{\left(q-1\right)}^{m-1}}\\ \times d^{m+\sigma -1}\sum _{k=0}^{d-1}\chi \left(k\right)B_{m+\sigma }\left(\frac{k+x}{d}\right).\end{array}$$

Then, because

$$\text{log}q=\text{log}\left(1+\left(q-1\right)\right)=\left(q-1\right)-\frac{{\left(q-1\right)}^2}{2}+\cdots =\left(q-1\right)+O\left({\left(q-1\right)}^2\right)$$

as q → 1, we find

$$\underset{q\to 1}{\text{lim}}\frac{{\left(\text{log}q\right)}^{m+\sigma -1}}{{\left(q-1\right)}^{m-1}}=\left\{\begin{array}{cc}1,\hfill & \sigma =0\hfill \\ 0,\hfill & \sigma \ge 1,\hfill \end{array}\right.$$

so

$$\underset{q\to 1}{\text{lim}}{\beta }_{m,\chi }\left(x,q\right)=d^{m-1}\sum _{k=0}^{d-1}\chi \left(k\right)B_m\left(\frac{k+x}{d}\right)=B_{m,\chi }\left(x\right),$$

where the B_m,χ(x) are the m th generalized Bernoulli polynomials (e.g., [14, 19]). This completes the proof.

Corollary 3.4. For any positive integer k, we have

$$\underset{q\to 1}{\text{lim}}{L}_q\left(1-k,x,\chi \right)=-\frac{1}{k}{B}_{k,x}\left(x\right).$$

Remark 3.5. The formula of Theorem 3.3 is slight extension of the result in [[26], Theorem 1].

Remark 3.6. From Theorem 2.7, the generalized Bernoulli polynomials B_m,χ(x) are defined by means of the following generating function [[27], p. 8]

$$\begin{array}{ll}\hfill F_{\chi }\left(t,x\right)& :=\underset{q\to 1}{\text{lim}}{F}_{q,\chi }\left(t,x\right)\\ =-t\sum _{a=1}^d\sum _{l=0}^{\infty }\chi \left(a+dl\right)e^{\left(a+dl\right)t}{e}^{xt}\\ =\sum _{a=1}^d\frac{\chi \left(a\right)t{e}^{\left(a+x\right)t}}{e^{dt}-1}\\ =\sum _{m=0}^{\infty }{B}_{m,\chi }\left(x\right)\frac{t^m}{m!}.\end{array}$$

Remark 3.7. If we substitute χ = χ⁰, the trivial character, in Definition 3.1 and Corollary 3.4, we can also define a q-analog of the Hurwitz zeta function

$$\zeta \left(s,x\right)=\sum _{m=0}^{\infty }\frac{1}{{\left(m+x\right)}^s}$$

by

$${\zeta }_q\left(s,x\right)=L_q\left(s,x,{\chi }^0\right)=\frac{1-q}{s-1}\sum _{m=0}^{\infty }\frac{q^{m+x}}{{\left[m+x\right]}_q^{s-1}}+\sum _{m=0}^{\infty }\frac{q^{2\left(m+x\right)}}{{\left[m+x\right]}_q^s}$$

and obtain the identity

$$\underset{q\to 1}{\text{lim}}{\zeta }_q\left(s,x\right)=\zeta \left(s,x\right)$$

for all s ≠ 1, as well as the formula

$$\underset{q\to 1}{\text{lim}}{\zeta }_q\left(1-k,x\right)=-\frac{1}{k}{B}_k\left(x\right)$$

for integers k ≥ 1 (cf. [11, 13, 19, 22, 24, 25]).