# Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials

• Lee-Chae Jang
• Taekyun Kim
• Young-Hee Kim
• Byungje Lee
Open Access
Research

## Abstract

In this paper, we consider the twisted Carlitz's q-Bernoulli numbers using p-adic q-integral on ℤ p . From the construction of the twisted Carlitz's q-Bernoulli numbers, we investigate some properties for the twisted Carlitz's q-Bernoulli numbers. Finally, we give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

## Keywords

q-Bernoulli numbers p-adic q-integral twisted

## 1. Introduction and preliminaries

Let p be a fixed prime number. Throughout this paper, ℤ p , and will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of , respectively. Let ℕ be the set of natural numbers, and let ℤ+ = ℕ ∪ {0}. Let ν p be the normalized exponential valuation of with . In this paper, we assume that with |1 - q| p < 1. The q-number is defined by . Note that limq → 1[x] q = x.

We say that f is a uniformly differentiable function at a point a ∈ ℤ p , and denote this property by fUD(ℤ p ), if the difference quotient has a limit f'(a) as (x, y) → (a, a). For fUD(ℤ p ), the p-adic q-integral on ℤ p , which is called the q-Volkenborn integral, is defined by Kim as follows:
(1)
In [2], Carlitz defined q-Bernoulli numbers, which are called the Carlitz's q-Bernoulli numbers, by
(2)

with the usual convention about replacing β n by βn, q.

In [2, 3], Carlitz also considered the expansion of q-Bernoulli numbers as follows:
(3)

with the usual convention about replacing (β(h)) n by .

Let be the cyclic group of order p n , and let (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]). Note that T p is a locally constant space.

For ξ ∈ T p , the twisted q-Bernoulli numbers are defined by
(4)
(see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]). From (4), we note that
(5)

with the usual convention about replacing by B n,ξ (see [17, 18, 19]). Recently, several authors have studied the twisted Bernoulli numbers and q-Bernoulli numbers in the area of number theory(see [17, 18, 19]).

In the viewpoint of (5), it seems to be interesting to investigate the twisted properties of (3). Using p-adic q-integral equation on ℤ p , we investigate the properties of the twisted q-Bernoulli numbers and polynomials related to q-Bernstein polynomials. From these properties, we derive some new identities for the twisted q-Bernoulli numbers and polynomials. Final purpose of this paper is to give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

## 2. On the twisted Carlitz 's q-Bernoulli numbers

In this section, we assume that n ∈ ℤ+, ξ ∈ T p and with |1 - q| p < 1.

Let us consider the n th twisted Carlitz's q-Bernoulli polynomials using p-adic q-integral on ℤ p as follows:
(6)

In the special case, x = 0, β n,ξ,q (0) = β n,ξ,q are called the n th twisted Carlitz's q-Bernoulli numbers.

From (6), we note that
(7)

Therefore, by (7), we obtain the following theorem.

Theorem 1. For n ∈ ℤ+, we have
Let F q, ξ (t, x) be the generating function of the twisted Carlitz's q-Bernoulli poly-nomials, which are given by
(8)

with the usual convention about replacing (β ξ,q (x)) n by β n,ξ,q (x).

By (8) and Theorem 1, we get
(9)
Let Fq,ξ(t, 0) = F q,ξ (t). Then, we have
(10)

Therefore, by (9) and (10), we obtain the following theorem.

Theorem 2. For n ∈ ℤ+, we have
From (6), we note that
(11)
with the usual convention about replacing (βξ,q) n by βn,ξ,q. By (11) and Theorem 2, we get
(12)
It is easy to show that
(13)

Therefore, by (13), we obtain the following theorem.

Theorem 3. For n ∈ ℤ+, we have
From Theorem 3, we can derive the following functional equation:
(14)

Therefore, by (14), we obtain the following corollary.

Corollary 4. Let. Then we have
By (11), we get that
(15)

Therefore, by (15), we obtain the following theorem.

Theorem 5. For n ∈ ℕ with n > 1, we have
By a simple calculation, we easily set
(16)
For n ∈ ℤ+ with n > 1, we have
(17)

Therefore, by (16) and (17), we obtain the following theorem.

Theorem 6. For n ∈ ℤ+with n > 1, we have
For x ∈ℤ p and n, k ∈ ℤ+, the p-adic q-Bernstein polynomials are given by
(18)

(see [8, 20]).

In [8], the q-Bernstein operator of order n is given by

Let f be continuous function on ℤ p . Then, the sequence converges uniformly to f on ℤ p (see [8]). If q is same version in (18), we cannot say that the sequence converges uniformly to f on ℤ p .

Let s ∈ ℕ with s ≥ 2. For n1, ..., n s , k ∈ ℤ+ with n1 + · · · + n s > sk + 1, we take the p-adic q-integral on ℤ p for the multiple product of q-Bernstein polynomials as follows:
(19)
and we also have
(20)

By comparing the coefficients on the both sides of (19) and (20), we obtain the following theorem.

Theorem 7. Let s ∈ ℕ with s ≥ 2. For n1, ..., n s , k ∈ ℤ+with n1 + ⋯ + n s > sk + 1, we have

## Notes

### Acknowledgements

The authors express their sincere gratitude to referees for their valuable suggestions and comments. This paper was supported by the research grant Kwangwoon University in 2011.

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© Kim et al; licensee Springer. 2011

## Authors and Affiliations

• Lee-Chae Jang
• 1
• Taekyun Kim
• 2
• Young-Hee Kim
• 2
• Byungje Lee
• 3
1. 1.Department of Mathematics and Computer ScienceKonkuk UniversityChungjuRepublic of Korea
2. 2.Division of General Education-MathematicsKwangwoon UniversitySeoulRepublic of Korea
3. 3.Department of Wireless Communications EngineeringKwangwoon UniversitySeoulRepublic of Korea