## Advances in Difference Equations

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# A note on Carlitz q-Bernoulli numbers and polynomials

Advances in Difference Equations20122012:44

DOI: 10.1186/1687-1847-2012-44

Received: 21 December 2011

Accepted: 13 April 2012

Published: 13 April 2012

## Abstract

In this article, we first aim to give simple proofs of known formulae for the generalized Carlitz q-Bernoulli polynomials βm,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.

### Keywords

Carlitz q-Bernoulli numbers Carlitz q-Bernoulli polynomials Dirichlet q-L- functions

## 1. Introduction

Let p be a fixed prime. We denote by p , p , and p the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of p , respectively. Let v p be the normalized exponential valuation of 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 , or a p-adic number q p . If q , one normally assumes |q| < 1. If q 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(ℤ/d{p}^{N}ℤ\right),\phantom{\rule{1em}{0ex}}{X}_{1}={ℤ}_{p},\phantom{\rule{2em}{0ex}}\\ \hfill {X}^{*}& =\underset{\underset{\left(a,p\right)=1}{0
(1.1)
where a lies in 0 ≤ a < dp N . We use the following notation:
${\left[x\right]}_{q}=\frac{1-{q}^{x}}{1-q}.$
(1.2)

Hence limq→1 [x] q = x for any x 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., [36]). 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,\phantom{\rule{1em}{0ex}}q{\left(q\beta \left(q\right)+1\right)}^{m}-{\beta }_{m}\left(q\right)=\left\{\begin{array}{cc}1\hfill & \mathsf{\text{if}}\phantom{\rule{0.3em}{0ex}}m=1\hfill \\ 0\hfill & \mathsf{\text{if}}\phantom{\rule{0.3em}{0ex}}m>1,\hfill \end{array}\right\$
(1.3)
with the usual convention of β i by 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 Gp,q(x) on p \ p .

In [11], Satoh constructed the generating function of the Carlitz q-Bernoulli numbers F q (t) in 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 p , see [11, p.347].

In [12], Kim defined the q-analog of the p-adic Haar distribution μHaar(a + p N p ) = 1/p N by
${\mu }_{q}\left(a+{p}^{N}{ℤ}_{p}\right)=\frac{{q}^{a}}{{\left[{p}^{N}\right]}_{q}}.$
(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 p by μ q , that is,
${\beta }_{m}\left(q\right)={\int }_{{ℤ}_{p}}{\left[a\right]}_{q}^{m}d{\mu }_{q}\left(a\right),$
(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 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 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 p with $0<{\left|1-q\right|}_{p}<{p}^{-\frac{1}{p-1}}$.

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 : p p , the p-adic q-integral on 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, 1416] 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 p with the weaker condition |1 - q| p < 1. Indeed, when |1 - q| p < 1 the function q x is uniformly differentiable and the space of uniformly differentiable functions p 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 p with 0 < |1 - q| p < 1 and x 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 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 p (see [[17], Lemma 3.1 (iii)]). Therefore, we obtain
$\begin{array}{ll}\hfill \underset{N\to \infty }{\text{lim}}\frac{1}{1-{q}^{{p}^{N}}}\sum _{a=0}^{{p}^{N}-1}{q}^{ax}& =\frac{1}{1-{q}^{x}}\underset{N\to \infty }{\text{lim}}\frac{{\left({q}^{{p}^{N}}\right)}^{x}-1}{{q}^{{p}^{N}}-1}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{1-{q}^{x}}\underset{N\to \infty }{\text{lim}}\frac{{\sum }_{m=1}^{\infty }\left(\begin{array}{c}x\hfill \\ m\hfill \end{array}\right){\left({q}^{{p}^{N}}-1\right)}^{m}}{{q}^{{p}^{N}}-1}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{1-{q}^{x}}\underset{N\to \infty }{\text{lim}}\sum _{m=0}^{\infty }\left(\begin{array}{c}x\hfill \\ m+1\hfill \end{array}\right){\left({q}^{{p}^{N}}-1\right)}^{m}\phantom{\rule{2em}{0ex}}\\ =\frac{x}{1-{q}^{x}}.\phantom{\rule{2em}{0ex}}\end{array}$

This completes the proof.

Definition 2.2 ([12, §2, p. 323]). Let χ be a primitive Dirichlet character with conductor d and let x p . For q p with 0 < |1 - q| p < 1 and an integer m ≥ 0, the generalized Carlitz q-Bernoulli polynomials βm,χ(x, q) are defined by
$\begin{array}{ll}\hfill {\beta }_{m,\chi }\left(x,q\right)& ={\int }_{X}\chi \left(a\right){\left[x+a\right]}_{q}^{m}d{\mu }_{q}\left(a\right)\phantom{\rule{2em}{0ex}}\\ =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[d{p}^{N}\right]}_{q}}\sum _{a=0}^{d{p}^{N}-1}\chi \left(a\right){\left[x+a\right]}_{q}^{m}{q}^{a}.\phantom{\rule{2em}{0ex}}\end{array}$
(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 βm,χ(x, q) in p .

Proposition 2.4. For q 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
$\begin{array}{ll}\hfill {\beta }_{m,\chi }\left(x,q\right)& =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[d\right]}_{q}}\frac{1}{{\left[{p}^{N}\right]}_{{q}^{d}}}\sum _{k=0}^{d-1}\sum _{a=0}^{{p}^{N}-1}\chi \left(k+da\right){\left[x+k+da\right]}_{q}^{m}{q}^{k+da}\phantom{\rule{2em}{0ex}}\\ =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left(1-q\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right)\frac{{q}^{k}}{1-{q}^{d{p}^{N}}}\sum _{q=0}^{{p}^{N}-1}{\left(1-{q}^{x+k+da}\right)}^{m}{q}^{da}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{{\left(1-q\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{k}\sum _{i=0}^{m}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^{i}{q}^{i\left(x+k\right)}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×\underset{N\to \infty }{\text{lim}}\frac{1}{1-{\left({q}^{d}\right)}^{{p}^{N}}}\sum _{a=0}^{{p}^{N}-1}{\left({q}^{d}\right)}^{a\left(i+1\right)}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{{\left(1-q\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{k}\sum _{i=0}^{m}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^{i}{q}^{i\left(x+k\right)}\frac{i+1}{1-{q}^{d\left(i+1\right)}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\mathsf{\text{(where}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{we}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{use}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{Lemma}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{2}}\mathsf{\text{.1).}}\phantom{\rule{2em}{0ex}}\end{array}$

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 βm,χ(0, q) in for m ≥ 0.

Lemma 2.6. Let χ be a primitive Dirichlet character with conductor d . Then for q 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 ≤ kd - 1 and a = 0,1, 2,..., we have the desired result.

We now consider the case:
$q\in \overline{ℚ}\cap {ℂ}_{p},\phantom{\rule{1em}{0ex}}0<\left|q\right|<1,\phantom{\rule{1em}{0ex}}0<{\left|1-q\right|}_{p}<1.$
(2.2)
For instance, if we set
$q=\frac{1}{1-pz}\in \overline{ℚ}\cap {ℂ}_{p}$

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

Let Fq,χ(t, x) be the generating function of βm,χ(x, q) defined in Definition 2.2. From Proposition 2.4, we have
$\begin{array}{ll}\hfill {F}_{q,\chi }\left(t,x\right)& =\sum _{m=0}^{\infty }{\beta }_{m,\chi }\left(x,q\right)\frac{{t}^{m}}{m!}\phantom{\rule{2em}{0ex}}\\ =\sum _{m=0}^{\infty }\left(\frac{1}{{\left(1-q\right)}^{m}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{k}\sum _{i=0}^{m}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^{i}{q}^{i\left(x+k\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_{q}}\right)\frac{{t}^{m}}{m!}\phantom{\rule{2em}{0ex}}\\ ={P}_{q,\chi }\left(t,x\right)+{Q}_{q,\chi }\left(t,x\right),\phantom{\rule{2em}{0ex}}\end{array}$
(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
$\begin{array}{ll}\hfill {P}_{q,\chi }\left(t,x\right)& =\sum _{m=0}^{\infty }\frac{1}{{\left(1-q\right)}^{m}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{k}\sum _{i=0}^{m}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^{i}{q}^{i\left(x+k\right)}\frac{i}{{\left[d\left(i+1\right)\right]}_{q}}\frac{{t}^{m}}{m!}\phantom{\rule{2em}{0ex}}\\ =\sum _{n=0}^{\infty }\frac{1}{{\left(1-q\right)}^{n}}\frac{{t}^{n}}{n!}\sum _{j=0}^{\infty }\frac{1}{{\left(q-1\right)}^{j}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{j\left(x+k\right)+k}\frac{j}{{\left[d\left(j+1\right)\right]}_{q}}\frac{{t}^{j}}{j!}\phantom{\rule{2em}{0ex}}\\ ={e}^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^{j}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{j\left(x+k\right)+k}\frac{j}{{\left[d\left(j+1\right)\right]}_{q}}\frac{{t}^{j}}{j!}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.4)
Moreover, (2.4) now becomes
$\begin{array}{ll}\hfill {P}_{q,\chi }\left(t,x\right)& ={e}^{\frac{t}{1-q}}\sum _{j=1}^{\infty }{\left(\frac{1}{q-1}\right)}^{j}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{j\left(x+k\right)+k}\frac{1}{{\left[d\left(j+1\right)\right]}_{q}}\frac{{t}^{j}}{\left(j-1\right)!}\phantom{\rule{2em}{0ex}}\\ ={e}^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^{j}{q}^{\left(j+1\right)x}\sum _{k=0}^{d-1}\chi \left(k\right)\frac{{q}^{k\left(j+2\right)}}{{{q}^{d}}^{\left(j+2\right)}-1}\frac{{t}^{j+1}}{j!}\phantom{\rule{2em}{0ex}}\\ =-t{e}^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^{j}{q}^{\left(j+1\right)x}\sum _{n=0}^{\infty }\chi \left(n\right){q}^{n\left(j+2\right)}\frac{{t}^{j}}{j!}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\mathsf{\text{(where}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{we}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{use}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{Lemma}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{2}}\mathsf{\text{.6)}}\phantom{\rule{2em}{0ex}}\\ =-t{e}^{\frac{t}{1-q}}\sum _{n=0}^{\infty }\chi \left(n\right){q}^{x+2n}\sum _{j=0}^{\infty }{\left(\frac{-{q}^{n+x}}{1-q}\right)}^{j}\frac{{t}^{j}}{j!}\phantom{\rule{2em}{0ex}}\\ =-t{e}^{\frac{t}{1-q}}\sum _{n=0}^{\infty }\chi \left(x\right){q}^{x+2n}{e}^{\frac{\left(-{{q}^{n}}^{+x}\right)t}{1-q}}\phantom{\rule{2em}{0ex}}\\ =-t\sum _{n=0}^{\infty }\chi \left(x\right){q}^{x+2n}{e}^{{\left[n+x\right]}_{q}t}\phantom{\rule{2em}{0ex}}\end{array}$
(2.5)
(cf. [13, 16, 20]). Similar arguments apply to the case Qq,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$q\in \overline{ℚ}\cap {ℂ}_{p},0<\left|q\right|<1,0<{\left|1-q\right|}_{p}<1$. Then the generalized Carlitz q-Bernoulli polynomials βm,χ(x, q) for m ≤ 0 is given by equating the coefficients of powers of t in the following generating function:
$\begin{array}{ll}\hfill {F}_{q,\chi }\left(t,x\right)& ={e}^{\frac{t}{1-q}}\sum _{j=0}^{\infty }{\left(\frac{1}{q-1}\right)}^{j-1}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{j\left(x+k\right)+k}\frac{j+1}{{q}^{d\left(j+1\right)}-1}\frac{{t}^{j}}{j!}\phantom{\rule{2em}{0ex}}\\ =\sum _{n=0}^{\infty }\chi \left(n\right){q}^{n}{e}^{{\left[n+x\right]}_{q}t}\left(1-q-{q}^{n+x}t\right).\phantom{\rule{2em}{0ex}}\end{array}$
(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
(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 . 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) × 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, 2125]).

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 , k ≥ 1, the limiting value limskL 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
$\begin{array}{ll}\hfill \underset{q\to 1}{\text{lim}}{\beta }_{k,\chi }\left(x,q\right)& =\underset{q\to 1}{\text{lim}}\frac{1}{{\left(1-q\right)}^{m}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{k}\sum _{i=0}^{m}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(-1\right)}^{i}{q}^{i\left(x+k\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_{q}}\phantom{\rule{2em}{0ex}}\\ ={B}_{k,\chi }\left(x\right),\phantom{\rule{2em}{0ex}}\end{array}$

where the Bk,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
(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 . From (2.3), (3.3), and (3.6), we obtain
$\begin{array}{ll}\hfill {\beta }_{m,\chi }\left(x,q\right)& =\frac{{q}^{-x}}{{\left(q-1\right)}^{m}}\sum _{k=0}^{d-1}\chi \left(k\right){q}^{k+x}\sum _{i=0}^{m}{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){q}^{i\left(k+x\right)}\frac{i+1}{{\left[d\left(i+1\right)\right]}_{q}}\phantom{\rule{2em}{0ex}}\\ =\frac{{q}^{-x}}{{\left(q-1\right)}^{m-1}}\sum _{k=0}^{d-1}\chi \left(k\right)\sum _{i=0}^{m}{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×{{e}^{d\left(i+1\right)}}^{\text{log}{q}^{\frac{\left(k+x\right)}{d}}}\frac{d\left(i+1\right)\text{log}q}{{e}^{d\left(i+1\right)\text{log}q-1}}\frac{1}{d\text{log}q}\phantom{\rule{2em}{0ex}}\\ =\frac{{q}^{-x}}{{\left(q-1\right)}^{m-1}}\sum _{n=0}^{\infty }\left(\sum _{i=0}^{m}{\left(-1\right)}^{m-i}\left(\begin{array}{c}m\hfill \\ i\hfill \end{array}\right){\left(i+1\right)}^{n}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×{d}^{n-1}\sum _{k=0}^{d-1}\chi \left(k\right){B}_{n}\left(\frac{k+x}{d}\right)\frac{{\left(\text{log}q\right)}^{n-1}}{n!}\phantom{\rule{2em}{0ex}}\\ ={q}^{-x}\frac{{\left(\text{log}q\right)}^{m-1}}{{\left(q-1\right)}^{m-1}}{d}^{m-1}\sum _{k=0}^{d-1}\chi \left(k\right){B}_{m}\left(\frac{k+x}{d}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{q}^{-x}\sum _{\sigma =1}^{\infty }\sum _{i=0}^{\sigma }\left(\begin{array}{c}m+\sigma \hfill \\ i\hfill \end{array}\right){d}_{m+\sigma -i}^{\left(m\right)}\frac{1}{\left(m+\sigma \right)!}\frac{{\left(\text{log}q\right)}^{m+\sigma -1}}{{\left(q-1\right)}^{m-1}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×{d}^{m+\sigma -1}\sum _{k=0}^{d-1}\chi \left(k\right){B}_{m+\sigma }\left(\frac{k+x}{d}\right).\phantom{\rule{2em}{0ex}}\end{array}$
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 Bm,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 Bm,χ(x) are defined by means of the following generating function [[27], p. 8]
$\begin{array}{ll}\hfill {F}_{\chi }\left(t,x\right)& :=\underset{q\to 1}{\text{lim}}{F}_{q,\chi }\left(t,x\right)\phantom{\rule{2em}{0ex}}\\ =-t\sum _{a=1}^{d}\sum _{l=0}^{\infty }\chi \left(a+dl\right){e}^{\left(a+dl\right)t}{e}^{xt}\phantom{\rule{2em}{0ex}}\\ =\sum _{a=1}^{d}\frac{\chi \left(a\right)t{e}^{\left(a+x\right)t}}{{e}^{dt}-1}\phantom{\rule{2em}{0ex}}\\ =\sum _{m=0}^{\infty }{B}_{m,\chi }\left(x\right)\frac{{t}^{m}}{m!}.\phantom{\rule{2em}{0ex}}\end{array}$
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]).

## Declarations

### Acknowledgements

This work was supported by the Kyungnam University Foundation Grant, 2012.

## Authors’ Affiliations

(1)
National Institute for Mathematical Sciences, Doryong-dong
(2)
Division of Cultural Education, Kyungnam University

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