q-Bernoulli numbers and q-Bernoulli polynomials revisited
© Ryoo et al; licensee Springer. 2011
Received: 26 February 2011
Accepted: 18 September 2011
Published: 18 September 2011
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
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.
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 ). 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. , some incorrect properties are revised.
In the special case, x = 0, B n,q (0) = B n,q are called the n th q-Bernoulli numbers.
Remark 1. Definition 1 is unreasonable, since it is not the generating function of q-Bernoulli numbers and polynomials.
By (6), we see that Definition 1 is unreasonable because we cannot derive Bernoulli numbers from Definition 1 for any q.
Therefore, by (4) and (6)-(9), we see that the following three theorems are incorrect.
In , 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.
In the special case, x = 0, β n,q (0) = β n,q are called the n th q-Bernoulli numbers.
with the usual convention about replacing by β n,q .
From (12), (14) and (16), Theorems 1-3 are revised by the following Theorems 1'-3'.
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 ).
Therefore, by (20) and (21), we obtain the following theorem.
Thus, by (22), we obtain the following theorem.
where x ≠ 0, -1, -2,....
where s ∈ ℂ, and x ≠ 0, -1, -2,....
Thus, we define q-zeta function as follows:
where x ≠ 0, -1, -2,....
where B n (x) are the n th ordinary Bernoulli polynomials.
The authors express their gratitude to the referee for his/her valuable comments.
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