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 [1–11]). 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 [4–9]).

It is known that the Bernoulli polynomials are defined as

$$\frac{t}{e^t-1}{e}^{xt}=\sum _{n=0}^{\infty }{B}_n\left(x\right)\frac{t^n}{n!}\mathsf{\text{for}}\left|t\right|<2\pi$$

(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:

$$B_0=1\mathsf{\text{and}}{\left(B+1\right)}^n-B_n=0\mathsf{\text{if}}n>0.$$

(1.2)

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

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

(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,

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

(2.1)

Note that

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

(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,

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

(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]),

$$\begin{array}{lll}\hfill D_q\left(t,x\right)& =\sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{t^n}{n!}& \hfill \text{(1)}\\ =B_{0,q}\left(x\right)+\sum _{n=1}^{\infty }{B}_{n,q}\left(x\right)\frac{t^n}{n!},& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(2.4)

and

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

(2.5)

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

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

(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

$$\begin{array}{lll}\hfill q{D}_q\left(t,1\right)-D_q\left(t\right)& =-t\sum _{y=0}^{\infty }{q}^{y+1}{e}^{{\left[1+y\right]}_{q}t}-t\sum _{y=0}^{\infty }{q}^y{e}^{{\left[y\right]}_{q}t}& \hfill \text{(1)}\\ =t,& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(2.7)

and

$$\begin{array}{lll}\hfill q{D}_q\left(t,1\right)-D_q\left(t\right)& =q\sum _{n=0}^{\infty }{B}_{n,q}\left(1\right)\frac{t^n}{n!}-\sum _{n=0}^{\infty }{B}_{n,q}\frac{t^n}{n!}& \hfill \text{(1)}\\ =\sum _{n=0}^{\infty }\left(q{B}_{n,q}\left(1\right)-B_{n,q}\right)\frac{t^n}{n!}.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(2.8)

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

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

(2.9)

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

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

(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,

$$F_q\left(t,x\right)=\frac{q-1}{logq}{e}^{\frac{1}{1-q}t}-t\sum _{m=0}^{\infty }{q}^{x+m}{e}^{{\left[x+m\right]}_{q}t}=\sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{t^n}{n!}.$$

(2.13)

Note that

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

(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,

$$F_q\left(t\right)=F_q\left(t,0\right)=\sum _{n=0}^{\infty }{B}_{n,q}\frac{t^n}{n!}.$$

(2.15)

By simple calculations, we get

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

(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.

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

(2.18)

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

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

(2.19)

Theorem 2.7* For n ∈ ℕ*, we have

$$B_{0,q}=\frac{q-1}{logq}\mathsf{\text{and}}{\left(q{B}_q+1\right)}^n-B_{n,q}=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \mathsf{\text{if}}n=1\hfill \\ \hfill 0\hfill & \hfill \mathsf{\text{if}}n>1\hfill \end{array}\right..$$

(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

$$\begin{array}{lll}\hfill \sum _{n=0}^{\infty }{B}_{n,q}\left(x\right)\frac{t^n}{n!}& =F_q\left(t,x\right)& \hfill \text{(1)}\\ =\frac{q-1}{logq}{e}^{\frac{1}{1-q}t}-t\sum _{m=0}^{\infty }{q}^{x+m}{e}^{{\left[x+m\right]}_{q}t}& \hfill \text{(2)}\\ =\frac{q-1}{logq}\sum _{n=0}^{\infty }\frac{1}{{\left(1-q\right)}^n}\frac{t^n}{n!}-\sum _{m=0}^{\infty }{q}^{x+m}\sum _{n=0}^{\infty }n{\left[x+m\right]}_q^{n-1}\frac{t^n}{n!}& \hfill \text{(3)}\\ =\sum _{n=0}^{\infty }\left(\frac{q-1}{logq}\frac{1}{{\left(1-q\right)}^n}-n\sum _{m=0}^{\infty }{q}^{x+m}{\left[x+m\right]}_q^{n-1}\right)\frac{t^n}{n!}& \hfill \text{(4)}\\ =\sum _{n=0}^{\infty }\left(-\frac{{\left(1-q\right)}^n}{logq}-\frac{n}{{\left(1-q\right)}^{n-1}}\sum _{m=0}^{\infty }{q}^{x+m}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{\left(x+m\right)l}\right)\frac{t^n}{n!}& \hfill \text{(5)}\\ =\sum _{n=0}^{\infty }\left(\frac{{\left(q-1\right)}^{1-n}}{logq}+\frac{n}{{\left(1-q\right)}^{n-1}}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+1}{q}^{\left(l+1\right)x}\frac{1}{1-q^{\left(l+1\right)}}\right)\frac{t^n}{n!}& \hfill \text{(6)}\\ =\sum _{n=0}^{\infty }\left(\frac{1}{{\left(1-q\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{q}^{lx}\frac{l}{{\left[l\right]}_q}\right)\frac{t^n}{n!}.& \hfill \text{(7)}\\ \hfill \text{(8) }\end{array}$$

(2.21)

By (2.21), we obtain the following theorem.

Theorem 2.9* For n ∈ ℕ*, we have

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

(2.22)

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