Theory and Modern Applications

# A new construction on the q-Bernoulli polynomials

## Abstract

This paper performs a further investigation on the q-Bernoulli polynomials and numbers given by Açikgöz et al. (Adv. Differ. Equ. 2010, 9, Article ID 951764) some incorrect properties are revised. It is pointed out that the definition concerning the q-Bernoulli polynomials and numbers is unreasonable. The purpose of this paper is to redefine the q-Bernoulli polynomials and numbers and correct its wrong properties and rebuild its theorems.

## 1 Introduction/Preliminaries

Many mathematicians have studied the q-Bernoulli, q-Euler polynomials and related topics (see [111]). It is worth that Açikgöz et al. [1] give a new approach to the q-Bernoulli polynomials and the q-Bernstein polynomials and show some properties. That is, Açikgöz et al. introduced a new generating function related the q-Bernoulli polynomials and gave a new construction of these polynomials related to the second kind Stirling numbers and the q-Bernstein polynomials in [1]. The purpose of this paper is to redefine a generating function related the q-Bernoulli polynomials and numbers and correct its wrong properties and rebuild its theorems.

In this paper, we assume that $q\left(\in ℂ\right)$ is indeterminate with |q| < 1. The q-number is defined by ${\left[x\right]}_{q}=\frac{{q}^{x}-1}{q-1}$ (see [49]).

It is known that the Bernoulli polynomials are defined as

$\frac{t}{{e}^{t}-1}{e}^{xt}=\sum _{n=0}^{\infty }{B}_{n}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{1em}{0ex}}|t|\phantom{\rule{2.77695pt}{0ex}}<2\pi$
(1.1)

and that B n (0) = B n are called the Bernoulli numbers.

The recurrence formula for the classical Bernoulli numbers B n is as follows:

${B}_{0}=1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{\left(B+1\right)}^{n}-{B}_{n}=0\phantom{\rule{1em}{0ex}}\mathsf{\text{if}}\phantom{\rule{1em}{0ex}}n>0.$
(1.2)

The q-extension of the following recurrence formula for the Bernoulli numbers is given by

${B}_{0,q}=1\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}q{\left(qB+1\right)}^{n}-{B}_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n=1\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>1\hfill \end{array}\right\$
(1.3)

with the usual convention of replacing ${B}_{q}^{n}$ by B n,q (see [2, 4]).

## 2 On the q-Bernoulli polynomials and numbers

In this section, we first recall the q-Bernoulli polynomials and numbers, then indicate the ambiguities on the Açikgöz et al. [1]'s definition for the q-Bernoulli polynomials and redefine it. Counter-examples show that some properties are incorrect. Specially, these examples show that the concept on the generating function of the q-Bernoulli polynomials is unreasonable.

Definition 2.1 (Açikgöz et al. [1]) For $q\in ℂ$ with |q| < 1, let us define the q-Bernoulli polynomials as follows,

${D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=-t\sum _{y=0}^{\infty }{q}^{y}{e}^{{\left[x+y\right]}_{q}t}=\sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{{t}^{n}}{n!}.$
(2.1)

Note that

$\underset{q\to 1}{lim}{D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\frac{t}{{e}^{t}-1}{e}^{xt}=\sum _{n=0}^{\infty }{B}_{n}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{1em}{0ex}}|t|\phantom{\rule{2.77695pt}{0ex}}<2\pi ,$
(2.2)

where B n (x) are the classical Bernoulli polynomials.

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

That is,

${D}_{q}\left(t\right)={D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}0\right)=-t\sum _{y=0}^{\infty }{q}^{y}{e}^{{\left[y\right]}_{q}t}=\sum _{n=0}^{\infty }{B}_{n,q}\frac{{t}^{n}}{n!}.$
(2.3)

Remark 2.2 Definition 2.1 (Açikgöz et al. [1]) is unreasonable, since it is not the generating functions of the q-Bernoulli polynomials and numbers. This can be seen the following counter-examples.

Counter-example 2.3 If we take t = 0 in (2.2) of Definition 2.1 (Açikgöz et al. [1]), then we have lim q→1 D q (0, x) = 0. But ${lim}_{t\to 0}\frac{t}{{e}^{t}-1}{e}^{xt}=1$ does not hold in the sense of Definition 2.1 (Açikgöz et al. [1]).

Counter-example 2.4 From (2.1) of Definition 2.1 (Açikgöz et al. [1]),

$\begin{array}{lll}\hfill {D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)& =\sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ ={B}_{0,q}\left(x\right)+\sum _{n=1}^{\infty }{B}_{n,q}\left(x\right)\frac{{t}^{n}}{n!},\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(2.4)

and

$\begin{array}{lll}\hfill {D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)& =-t\sum _{y=0}^{\infty }{q}^{y}{e}^{{\left[x+y\right]}_{q}t}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =-t\sum _{y=0}^{\infty }{q}^{y}\sum _{n=0}^{\infty }{\left[x+y\right]}_{q}^{n}\frac{{t}^{n}}{n!}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\sum _{n=0}^{\infty }\left(-\frac{1}{{\left(1-q\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{lx}\sum _{y=0}^{\infty }{q}^{\left(l+1\right)y}\right)\frac{{t}^{n+1}}{n!}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\sum _{n=0}^{\infty }\left(-\frac{n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{lx}\frac{l}{1-{q}^{l+1}}\right)\frac{{t}^{n}}{n!}.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
(2.5)

Comparing these identities (2.4) and (2.5), we obtain

${B}_{0,q}\left(x\right)=0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{B}_{n,q}\left(x\right)=-\frac{n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{lx}\frac{l}{1-{q}^{l+1}}.$
(2.6)

This cannot satisfy some well-known results related the Bernoulli polynomials and numbers. For example, B 0 = 1.

Counter-example 2.5 From Definition 2.1 (Açikgöz et al. [1]), we note that

$\begin{array}{lll}\hfill q{D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}1\right)-{D}_{q}\left(t\right)& =-t\sum _{y=0}^{\infty }{q}^{y+1}{e}^{{\left[1+y\right]}_{q}t}-t\sum _{y=0}^{\infty }{q}^{y}{e}^{{\left[y\right]}_{q}t}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =t,\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(2.7)

and

$\begin{array}{lll}\hfill q{D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}1\right)-{D}_{q}\left(t\right)& =q\sum _{n=0}^{\infty }{B}_{n,q}\left(1\right)\frac{{t}^{n}}{n!}-\sum _{n=0}^{\infty }{B}_{n,q}\frac{{t}^{n}}{n!}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\sum _{n=0}^{\infty }\left(q{B}_{n,q}\left(1\right)-{B}_{n,q}\right)\frac{{t}^{n}}{n!}.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(2.8)

From (2.7) and (2.8), we can easily derive that

${B}_{n,q}=0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}q{B}_{n,q}\left(1\right)-{B}_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n=1\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>1\hfill \end{array}\right\.$
(2.9)

From (2.1) of Definition 2.1 (Açikgöz et al. [1]),

$\begin{array}{lll}\hfill \sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{{t}^{n}}{n!}& ={D}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =-t\sum _{y=0}^{\infty }{q}^{y}{e}^{{\left[x+y\right]}_{q}t}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ ={e}^{{\left[x\right]}_{q}t}\frac{1}{{q}^{x}}{D}_{q}\left(t{q}^{x}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\left(\sum _{l=0}^{\infty }\frac{{\left[x\right]}_{q}^{l}{t}^{l}}{l!}\right)×\left(\sum _{m=0}^{\infty }{B}_{m,q}\frac{{q}^{\left(m-1\right)x}{t}^{m}}{m!}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ =\sum _{n=0}^{\infty }\left(\sum _{m=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right){B}_{m,q}\phantom{\rule{0.3em}{0ex}}{q}^{\left(m-1\right)x}{\left[x\right]}_{q}^{n-m}\right)\frac{{t}^{n}}{n!}.\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\\ \hfill \text{(6)}\end{array}$
(2.10)

If we compare the coefficients on the both sides in (2.10),

${B}_{n,q}\left(x\right)=\sum _{m=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill m\hfill \end{array}\right){B}_{m,q}{q}^{\left(m-1\right)x}{\left[x\right]}_{q}^{n-m}.$
(2.11)

From (2.9) and (2.11),

${B}_{0,q}\left(x\right)=\frac{1}{{q}^{x}}{B}_{0,q}=0.$
(2.12)

However, these are also incorrect.

Next, we redefine the q-Bernoulli polynomials and numbers.

Definition 2.6 For $q\in ℂ$ with |q| < 1, let us define the q-Bernoulli polynomials B n,q (x) as follows,

${F}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\frac{q-1}{logq}{e}^{\frac{1}{1-q}t}-t\sum _{m=0}^{\infty }{q}^{x+m}{e}^{{\left[x+m\right]}_{q}t}=\sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{{t}^{n}}{n!}.$
(2.13)

Note that

$\underset{q\to 1}{lim}{F}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\frac{t}{{e}^{t}-1}{e}^{xt}=\sum _{n=0}^{\infty }{B}_{n}\left(x\right)\frac{{t}^{n}}{n!}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}|t|\phantom{\rule{2.77695pt}{0ex}}<2\pi ,$
(2.14)

where B n (x) are the classical Bernoulli polynomials.

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

${F}_{q}\left(t\right)={F}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}0\right)=\sum _{n=0}^{\infty }{B}_{n,q}\frac{{t}^{n}}{n!}.$
(2.15)

By simple calculations, we get

$\begin{array}{lll}\hfill \sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{{t}^{n}}{n!}& ={F}_{q}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ ={e}^{{\left[x\right]}_{q}t}{F}_{q}\left({q}^{x}t\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\left(\sum _{m=0}^{\infty }\frac{{\left[x\right]}_{q}^{m}{t}^{m}}{m!}\right)×\left(\sum _{l=0}^{\infty }{B}_{l,q}\frac{{q}^{lx}{t}^{l}}{l!}\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =\sum _{n=0}^{\infty }\left(\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){B}_{l,q}{q}^{lx}{\left[x\right]}_{q}^{n-l}\right)\frac{{t}^{n}}{n!}.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
(2.16)

Comparing the coefficients on the both sides in (2.16), we obtain

${B}_{n,q}\left(x\right)=\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){B}_{l,q}{q}^{lx}{\left[x\right]}_{q}^{n-l}.$
(2.17)

From (2.13) and (2.15), we derive the following equation.

${B}_{0,q}=\frac{q-1}{logq}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{B}_{n,q}\left(1\right)-{B}_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n=1\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>1\hfill \end{array}\right\.$
(2.18)

By (2.17) and (2.18), we can see that

${B}_{0,q}=\frac{q-1}{logq}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){B}_{l,q}{q}^{l}-{B}_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n=1\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>1\hfill \end{array}\right\.$
(2.19)

Theorem 2.7* For n *, we have

${B}_{0,q}=\frac{q-1}{logq}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{\left(q{B}_{q}+1\right)}^{n}-{B}_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n=1\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>1\hfill \end{array}\right\.$
(2.20)

with the usual convention of replacing ${B}_{q}^{n}$ by B n,q .

Remark 2.8 Theorem 2.7* is a revised theorem of Theorem 2.1 in [1].

From (2. 13), we have

(2.21)

By (2.21), we obtain the following theorem.

Theorem 2.9* For n *, we have

${B}_{0,q}=\frac{q-1}{logq}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{B}_{n,q}\left(x\right)=\frac{1}{{\left(1-q\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{lx}\frac{l}{{\left[l\right]}_{q}}.$
(2.22)

Remark 2.10 Theorem 2.9* is a revised theorem of Theorem 2.3 in [1].

## References

1. Açikgöz M, Erdal D, Araci S: A new approach to q -Bernoulli numbers and q -Bernoulli polynomials related to q -Bernstein polynomials. Adv Differ Equ 2010., 9: Article ID 951764

2. Carlitz L: q-Bernoulli numbers and polynomials. Duke Math J 1948, 15: 987-1000. 10.1215/S0012-7094-48-01588-9

3. Kac V, Cheung P: Quantum Calculus, Universitext. Springer, New York; 2001.

4. Kim T: A new approach to q -zeta function. Adv Stud Contemp Math 2005,11(2):157-162.

5. Kim T: q -Volkenborn integration. Russ. J Math Phys 2002,9(3):288-299.

6. Kim T: On p -adic q - L -functions and sums of powers. Discret Math 2002,252(1-3):179-187. 10.1016/S0012-365X(01)00293-X

7. Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ J Math Phys 2008,15(1):51-57.

8. Kim T: Power series and asymptotic series associated with the q -analog of the two-variable p -adic L -function. Russ J Math Phys 2005,12(2):186-196.

9. Kim T: Analytic continuation of multiple q -zeta functions and their values at negative integers. Russ J Math Phys 2004,11(1):71-76.

10. Kim T: On explicit formulas of p -adic q - L -functions. Kyushu J Math 1994,48(1):73-86. 10.2206/kyushujm.48.73

11. Kim T, Kim HS: Remark on p -adic q -Bernoulli numbers, Algebraic number theory (Hapcheon/Saga, 1996). Adv Stud Contemp Math Pusan 1999, 1: 127-136.

## Acknowledgements

The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestions.

This research was supported by Kyungpook National University Research Fund, 2010.

## Author information

Authors

### Corresponding author

Correspondence to Seog-Hoon Rim.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

Coresponding author raised the problem and make a sequence to appoach the problem. AB carried out the q-Bernoulli poynomials studies, participated in the making new construction of the q-Bernoulli numbers. EJM carried out the calculation of [1]. JHJ participated in the sequence alignment. SJL performed the correction problem. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and permissions

Rim, SH., Bayad, A., Moon, EJ. et al. A new construction on the q-Bernoulli polynomials. Adv Differ Equ 2011, 34 (2011). https://doi.org/10.1186/1687-1847-2011-34