q-Bernoulli numbers and q-Bernoulli polynomials revisited

Abstract

This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010), some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994) (see Equation 9), some new generating functions for the q-Bernoulli numbers and polynomials are shown.

Mathematics Subject Classification (2000) 11B68, 11S40, 11S80

1. Introduction

As well-known definition, the Bernoulli polynomials are given by

$$\frac{t}{e^t-1}{e}^{xt}=e^{B\left(x\right)t}=\sum _{n=0}^{\infty }{B}_n\left(x\right)\frac{t^n}{n!},$$

(see [1–4]),

with usual convention about replacing B ⁿ(x) by B _n (x). In the special case, x = 0, B _n (0) = B _n are called the n th Bernoulli numbers.

Let us assume that q ∈ ℂ with |q| < 1 as an indeterminate. The q-number is defined by

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

(see [1–6]).

Note that lim _q→1[x] _q = x.

Since Carlitz brought out the concept of the q-extension of Bernoulli numbers and polynomials, many mathematicians have studied q-Bernoulli numbers and q-Bernoulli polynomials (see [1, 7, 5, 6, 8–12]). Recently, Acikgöz, Erdal, and Araci have studied to a new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials (see [7]). But, their generating function is unreasonable. The wrong properties are indicated by some counter-examples, and they are corrected.

It is point out that Acikgöz, Erdal and Araci's generating function for q-Bernoulli numbers and polynomials is unreasonable by counter examples, then the new generating function for the q-Bernoulli numbers and polynomials are given.

2. q-Bernoulli numbers and q-Bernoulli polynomials revisited

In this section, we perform a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. [7], some incorrect properties are revised.

Definition 1 (Acikgöz et al. [7]). For q ∈ ℂ with |q| < 1, let us define q-Bernoulli polynomials as follows:

$$D_q\left(t,x\right)=-t\sum _{y=0}^{\infty }{q}^y{e}^{{\left[x+y\right]}_{q}t}=\sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{t^n}{n!},where|t+logq|<2\pi .$$

(1)

In the special case, x = 0, B _n,q (0) = B _n,q are called the n th q-Bernoulli numbers.

Let D _q (t, 0) = D _q (t). Then

$$D_q\left(t\right)=-t\sum _{y=0}^{\infty }{q}^y{e}^{{\left[y\right]}_{q}t}=\sum _{n=0}^{\infty }{B}_{n,q}\frac{t^n}{n!}.$$

(2)

Remark 1. Definition 1 is unreasonable, since it is not the generating function of q-Bernoulli numbers and polynomials.

Indeed, by (2), we get

$$\begin{array}{lll}\hfill D_q\left(t,x\right)& =-t\sum _{y=0}^{\infty }{q}^y{e}^{{\left[x+y\right]}_{q}t}=-t\sum _{y=0}^{\infty }{q}^y{e}^{{\left[x\right]}_{q}t}{e}^{q^x{\left[y\right]}_{q}t}& \hfill \text{(1)}\\ =\left(-\frac{q^{x}t}{q^x}\sum _{y=0}^{\infty }{q}^y{e}^{q^x{\left[y\right]}_{q}t}\right)e^{{\left[x\right]}_{q}t}& \hfill \text{(2)}\\ =\frac{1}{q^x}{e}^{{\left[x\right]}_{q}t}{D}_q\left(q^{x}t\right)& \hfill \text{(3)}\\ =\left(\sum _{m=0}^{\infty }\frac{{\left[x\right]}_q^m}{m!}{t}^m\right)\left(\sum _{l=0}^{\infty }\frac{q^{\left(l-1\right)x}{B}_{l,q}}{l!}{t}^l\right)& \hfill \text{(4)}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_q^{n-l}{q}^{\left(l-1\right)x}{B}_{l,q}\right)\frac{t^n}{n!}.& \hfill \text{(5)}\\ \hfill \text{(6)}\end{array}$$

(3)

By comparing the coefficients on the both sides of (1) and (3), we obtain the following equation

$$B_{n,q}\left(x\right)=\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_q^{n-l}{q}^{\left(l-1\right)x}{B}_{l,q}.$$

(4)

From (1), we note that

$$\begin{array}{lll}\hfill D_q\left(t,x\right)& =-t\sum _{y=0}^{\infty }{q}^y{e}^{{\left[x+y\right]}_{q}t}& \hfill \text{(1)}\\ =\sum _{n=0}^{\infty }\left(-t\sum _{y=0}^{\infty }{q}^y{\left[x+y\right]}_q^n\right)\frac{t^n}{n!}& \hfill \text{(2)}\\ =-\sum _{n=0}^{\infty }\left(\frac{n+1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\sum _{y=0}^{\infty }{q}^{\left(l+1\right)y}\right)\frac{t^{n+1}}{\left(n+1\right)!}& \hfill \text{(3)}\\ =\sum _{n=1}^{\infty }\left(\frac{-n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\left(\frac{1}{1-q^{l+1}}\right)\right)\frac{t^n}{n!}.& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$$

(5)

By comparing the coefficients on the both sides of (1) and (5), we obtain the following equation

$$\begin{array}{c}{B}_{0,q}=0,\\ B_{n,q}=\frac{-n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\left(\frac{1}{1-q^{l+1}}\right)ifn>0.\end{array}$$

(6)

By (6), we see that Definition 1 is unreasonable because we cannot derive Bernoulli numbers from Definition 1 for any q.

In particular, by (1) and (2), we get

$$q{D}_q\left(t,1\right)-D_q\left(t\right)=t.$$

(7)

Thus, by (7), we have

$$q{B}_{n,q}\left(1\right)-B_{n,q}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill ifn=1,\hfill \\ \hfill 0,\hfill & \hfill ifn>1,\hfill \end{array}\right.$$

(8)

and

$$B_{n,q}\left(1\right)=\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{l-1}{B}_{l,q}.$$

(9)

Therefore, by (4) and (6)-(9), we see that the following three theorems are incorrect.

Theorem 1 (Acikgöz et al. [7]). For n ∈ ℕ*, one has

$$B_{0,q}=1,q{\left(qB+1\right)}^n-B_{n,q}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill ifn=0,\hfill \\ \hfill 0,\hfill & \hfill ifn>0.\hfill \end{array}\right.$$

Theorem 2 (Acikgöz et al. [7]). For n ∈ ℕ*, one has

$$B_{n,q}\left(x\right)=\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{lx}{B}_{l,q}{\left[x\right]}_q^{n-l}.$$

Theorem 3 (Acikgöz et al. [7]). For n ∈ ℕ*, one has

$$B_{n,q}\left(x\right)=\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\frac{l+1}{{\left[l+1\right]}_q}.$$

In [7], Acikgöz, Erdal and Araci derived some results by using Theorems 1-3. Hence, the other results are incorrect.

Now, we redefine the generating function of q-Bernoulli numbers and polynomials and correct its wrong properties, and rebuild the theorems of q-Bernoulli numbers and polynomials.

Redefinition 1. For q ∈ ℂ with |q| < 1, let us define q-Bernoulli polynomials as follows:

$$\begin{array}{lll}\hfill F_q\left(t,x\right)& =-t\sum _{m=0}^{\infty }{q}^{2m+x}{e}^{{\left[x+m\right]}_{q}t}+\left(1-q\right)\sum _{m=0}^{\infty }{q}^m{e}^{{\left[x+m\right]}_{q}t}& \hfill \text{(1)}\\ =\sum _{n=0}^{\infty }{\beta }_{n,q}\left(x\right)\frac{t^n}{n!},where|t+logq|<2\pi .& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(10)

In the special case, x = 0, β _n,q (0) = β _n,q are called the n th q-Bernoulli numbers.

Let F _q (t, 0) = F _q (t). Then we have

$$\begin{array}{lll}\hfill F_q\left(t\right)& =\sum _{n=0}^{\infty }{\beta }_{n,q}\frac{t^n}{n!}& \hfill \text{(1)}\\ =-t\sum _{m=0}^{\infty }{q}^{2m}{e}^{{\left[m\right]}_{q}t}+\left(1-q\right)\sum _{m=0}^{\infty }{q}^m{e}^{{\left[m\right]}_{q}t}.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(11)

By (10), we get

$$\begin{array}{lll}\hfill {\beta }_{n,q}\left(x\right)& =-n\sum _{m=0}^{\infty }{q}^{2m+x}{\left[x+m\right]}_q^{n-1}+\left(1-q\right)\sum _{m=0}^{\infty }{q}^m{\left[x+m\right]}_q^n& \hfill \text{(1) }\\ =\frac{-n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{\left(l+1\right)x}}{\left(1-q^{l+2}\right)}+\left(1-q\right)\sum _{m=0}^{\infty }{q}^m{\left[x+m\right]}_q^n& \hfill \text{(2) }\\ =\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\frac{l+1}{{\left[l+1\right]}_q}.& \hfill \text{(3) }\\ \hfill \text{(4) }\end{array}$$

(12)

By (10) and (11), we get

$$\begin{array}{lll}\hfill F_q\left(t,x\right)& =e^{{\left[x\right]}_{q}t}{F}_q\left(q^{x}t\right)& \hfill \text{(1)}\\ =\left(\sum _{m=0}^{\infty }{\left[x\right]}_q^m\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\infty }\frac{{\beta }_{l,q}}{l!}{q}^{lx}{t}^l\right)& \hfill \text{(2)}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\frac{q^{lx}{\beta }_{l,q}{\left[x\right]}_q^{n-l}n!}{l!\left(n-l\right)!}\right)\frac{t^n}{n!}& \hfill \text{(3)}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{lx}{\beta }_{l,q}{\left[x\right]}_q^{n-l}\right)\frac{t^n}{n!}.& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$$

(13)

Thus, by (12) and (13), we have

$$\begin{array}{lll}\hfill {\beta }_{n,q}\left(x\right)& =\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{lx}{\beta }_{l,q}{\left[x\right]}_q^{n-l}& \hfill \text{(1)}\\ =-n\sum _{m=0}^{\infty }{q}^m{\left[x+m\right]}_q^{n-1}+\left(1-q\right)\left(n+1\right)\sum _{m=0}^{\infty }{q}^m{\left[x+m\right]}_q^n.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(14)

From (10) and (11), we can derive the following equation:

$$q{F}_q\left(t,1\right)-F_q\left(t\right)=t+\left(q-1\right).$$

(15)

By (15), we get

$$q{\beta }_{n,q}\left(1\right)-{\beta }_{n,q}=\left\{\begin{array}{cc}\hfill q-1,\hfill & \hfill ifn=0,\hfill \\ \hfill 1,\hfill & \hfill ifn=1,\hfill \\ \hfill 0\hfill & \hfill ifn>1.\hfill \end{array}\right.$$

(16)

Therefore, by (14) and (15), we obtain

$${\beta }_{0,q}=1,q{\left(q{\beta }_q+1\right)}^n-{\beta }_{n,q}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill ifn=1,\hfill \\ \hfill 0\hfill & \hfill ifn>1,\hfill \end{array}\right.$$

(17)

with the usual convention about replacing ${\beta }_q^n$ by β _n,q .

From (12), (14) and (16), Theorems 1-3 are revised by the following Theorems 1'-3'.

Theorem 1'. For n ∈ ℤ₊, we have

$${\beta }_{0,q}=1,andq{\left(q{\beta }_q+1\right)}^n-{\beta }_{n,q}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill ifn=1,\hfill \\ \hfill 0\hfill & \hfill ifn>1.\hfill \end{array}\right.$$

Theorem 2'. For n ∈ ℤ₊, we have

$${\beta }_{n,q}\left(x\right)=\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{lx}{\beta }_{l,q}{\left[x\right]}_q^{n-l}.$$

Theorem 3'. For n ∈ ℤ₊, we have

$${\beta }_{n,q}\left(x\right)=\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\frac{l+1}{{\left[l+1\right]}_q}.$$

From (10), we note that

$$F_q\left(t,x\right)=\frac{1}{{\left[d\right]}_q}\sum _{a=0}^{d-1}{q}^a{F}_{q^d}\left({\left[d\right]}_{q}t,\frac{x+a}{d}\right),d\in \mathbb{N}.$$

(18)

Thus, by (10) and (18), we have

$${\beta }_{n,q}\left(x\right)={\left[d\right]}_q^{n-1}\sum _{a=0}^{d-1}{q}^a{\beta }_{n,q^d}\left(\frac{x+a}{d}\right),n\in {\mathbb{Z}}_{+}.$$

For d ∈ ℕ, let χ be Dirichlet's character with conductor d. Then, we consider the generalized q-Bernoulli polynomials attached to χ as follows:

$$\begin{array}{lll}\hfill F_{q,\chi }\left(t,x\right)& =-t\sum _{m=0}^{\infty }\chi \left(m\right)q^{2m+x}{e}^{{\left[x+m\right]}_{q}t}+\left(1-q\right)\sum _{m=0}^{\infty }\chi \left(m\right)q^m{e}^{{\left[x+m\right]}_{q}t}& \hfill \text{(1)}\\ =\sum _{n=0}^{\infty }{\beta }_{n,\chi ,q}\left(x\right)\frac{t^n}{n!}.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

In the special case, x = 0, β _n,χ,q (0) = β _n,χ,q are called the n th generalized Carlitz q-Bernoulli numbers attached to χ (see [8]).

Let F _q,χ (t, 0) = F _q,χ (t). Then we have

$$\begin{array}{lll}\hfill F_{q,\chi }\left(t\right)& =-t\sum _{m=0}^{\infty }\chi \left(m\right)q^{2m}{e}^{{\left[m\right]}_{q}t}+\left(1-q\right)\sum _{m=0}^{\infty }\chi \left(m\right)q^m{e}^{{\left[m\right]}_{q}t}& \hfill \text{(1)}\\ =\sum _{n=0}^{\infty }{\beta }_{n,\chi ,q}\frac{t^n}{n!}.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(20)

From (20), we note that

$$\begin{array}{lll}\hfill {\beta }_{n,\chi ,q}& =-n\sum _{m=0}^{\infty }{q}^{2m}\chi \left(m\right){\left[m\right]}_q^{n-1}+\left(1-q\right)\sum _{m=0}^{\infty }{q}^m\chi \left(m\right){\left[m\right]}_q^n& \hfill \text{(1)}\\ =-n\sum _{a=0}^{d-1}\sum _{m=0}^{\infty }{q}^{2a+2dm}\chi \left(a+dm\right){\left[a+dm\right]}_q^{n-1}& \hfill \text{(2)}\\ +\sum _{a=0}^{d-1}\sum _{m=0}^{\infty }{q}^{a+dm}\chi \left(a+dm\right){\left[a+dm\right]}_q^n& \hfill \text{(3)}\\ =\sum _{a=0}^{d-1}\chi \left(a\right)q^a\left(\frac{-n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{\left(l+1\right)a}}{\left(1-q^{d\left(l+2\right)}\right)}\right)& \hfill \text{(4)}\\ +\left(1-q\right)\sum _{a=0}^{d-1}\chi \left(a\right)q^a\left(\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{la}}{\left(1-q^{d\left(l+1\right)}\right)}\right)& \hfill \text{(5)}\\ =\sum _{a=0}^{d-1}\chi \left(a\right)q^a\left(\frac{-n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{\left(l+1\right)a}}{\left(1-q^{d\left(l+2\right)}\right)}\right)& \hfill \text{(6)}\\ +\sum _{a=0}^{d-1}\chi \left(a\right)q^a\left(\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{la}}{\left(1-q^{d\left(l+1\right)}\right)}\right)& \hfill \text{(7)}\\ =\sum _{a=0}^{d-1}\chi \left(a\right)q^a\left(\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{la}l}{\left(1-q^{d\left(l+1\right)}\right)}\right)& \hfill \text{(8)}\\ +\sum _{a=0}^{d-1}\chi \left(a\right)q^a\left(\frac{1}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)\frac{{\left(-1\right)}^l{q}^{la}}{\left(1-q^{d\left(l+1\right)}\right)}\right)& \hfill \text{(9)}\\ =\sum _{a=0}^{d-1}\chi \left(a\right)q^a\frac{1-q}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{la}\left(\frac{l+1}{1-q^{d\left(l+1\right)}}\right).& \hfill \text{(10)}\\ \hfill \text{(11)}\end{array}$$

Therefore, by (20) and (21), we obtain the following theorem.

Theorem 4. For n ∈ ℤ₊, we have

$$\begin{array}{lll}\hfill {\beta }_{n,\chi ,q}& =\sum _{a=0}^{d-1}\chi \left(a\right)q^a\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{la}\frac{l+1}{{\left[d\left(l+1\right)\right]}_q}& \hfill \text{(1)}\\ =-n\sum _{m=0}^{\infty }\chi \left(m\right)q^m{\left[m\right]}_q^{n-1}+\left(1-q\right)\left(1+n\right)\sum _{m=0}^{\infty }\chi \left(m\right)q^m{\left[m\right]}_q^n,& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

and

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

From (19), we note that

$$F_{q,\chi }\left(t,x\right)=\frac{1}{{\left[d\right]}_q}\sum _{a=0}^{d-1}\chi \left(a\right)q^a{F}_{q^d}\left({\left[d\right]}_{q}t,\frac{x+a}{d}\right).$$

(22)

Thus, by (22), we obtain the following theorem.

Theorem 5. For n ∈ ℤ₊, we have

$${\beta }_{n,\chi ,q}\left(x\right)={\left[d\right]}_q^{n-1}\sum _{a=0}^{d-1}\chi \left(a\right)q^a{\beta }_{n,q^d}\left(\frac{x+a}{d}\right).$$

For s ∈ ℂ, we now consider the Mellin transform for F _q (t, x) as follows:

$$\frac{1}{\Gamma \left(s\right)}\underset{0}{\overset{\infty }{\int }}{F}_q\left(-t,x\right)t^{s-2}dt=\sum _{m=0}^{\infty }\frac{q^{2m+x}}{{\left[m+x\right]}_q^s}+\frac{1-q}{s-1}\sum _{m=0}^{\infty }\frac{q^m}{{\left[m+x\right]}_q^{s-1}},$$

(23)

where x ≠ 0, -1, -2,....

From (23), we note that

$$\begin{array}{c}\frac{1}{\Gamma \left(s\right)}\underset{0}{\overset{\infty }{\int }}{F}_q\left(-t,x\right)t^{s-2}dt\\ =\sum _{m=0}^{\infty }\frac{q^m}{{\left[m+x\right]}_q^s}+\left(1-q\right)\left(\frac{2-s}{s-1}\right)\sum _{m=0}^{\infty }\frac{q^m}{{\left[m+x\right]}_q^{s-1}},\end{array}$$

(24)

where s ∈ ℂ, and x ≠ 0, -1, -2,....

Thus, we define q-zeta function as follows:

Definition 2. For s ∈ ℂ, q-zeta function is defined by

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

where x ≠ 0, -1, -2,....

By (24) and Definition 2, we note that

$${\zeta }_q\left(1-n,x\right)={(-1)}^{n-1}\frac{{\beta }_{n,q}\left(x\right)}{n},n\in \mathbb{N}.$$

Note that

$$\underset{q\to 1}{lim}{\zeta }_q\left(1-n,x\right)=-\frac{B_n\left(x\right)}{n},$$

where B _n (x) are the n th ordinary Bernoulli polynomials.