A new construction on the q-Bernoulli polynomials
Advances in Difference Equations volume 2011, Article number: 34 (2011)
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.
Many mathematicians have studied the q-Bernoulli, q-Euler polynomials and related topics (see [1–11]). It is worth that Açikgöz et al.  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 . 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.
It is known that the Bernoulli polynomials are defined as
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:
The q-extension of the following recurrence formula for the Bernoulli numbers is given by
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. '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. ) For with |q| < 1, let us define the q-Bernoulli polynomials as follows,
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.
Remark 2.2 Definition 2.1 (Açikgöz et al. ) 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 2.4 From (2.1) of Definition 2.1 (Açikgöz et al. ),
Comparing these identities (2.4) and (2.5), we obtain
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. ), we note that
From (2.7) and (2.8), we can easily derive that
From (2.1) of Definition 2.1 (Açikgöz et al. ),
If we compare the coefficients on the both sides in (2.10),
From (2.9) and (2.11),
However, these are also incorrect.
Next, we redefine the q-Bernoulli polynomials and numbers.
Definition 2.6 For with |q| < 1, let us define the q-Bernoulli polynomials B n,q (x) as follows,
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,
By simple calculations, we get
Comparing the coefficients on the both sides in (2.16), we obtain
From (2.13) and (2.15), we derive the following equation.
By (2.17) and (2.18), we can see that
Theorem 2.7* For n ∈ ℕ*, we have
with the usual convention of replacing by B n,q .
Remark 2.8 Theorem 2.7* is a revised theorem of Theorem 2.1 in .
From (2. 13), we have
By (2.21), we obtain the following theorem.
Theorem 2.9* For n ∈ ℕ*, we have
Remark 2.10 Theorem 2.9* is a revised theorem of Theorem 2.3 in .
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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.
The authors declare that they have no competing interests.
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 . JHJ participated in the sequence alignment. SJL performed the correction problem. All authors read and approved the final manuscript.
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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
- Generate Function
- Functional Equation
- Related Topic
- Wrong Property
- Recurrence Formula