Some Identities on the q-Bernoulli Numbers and Polynomials with Weight 0

and Applied Analysis 3 where n, k ∈ Z see 1, 9, 10 . For n, k ∈ Z , the p-adic Bernstein polynomials of degree n are defined by Bk,n x k x k 1 − x n−k for x ∈ Zp, see 1, 10, 11 . In this paper, we consider Bernstein polynomials to express the p-adic q-integral on Zp and investigate some interesting identities of Bernstein polynomials associated with the q-Bernoulli numbers and polynomials with weight 0 by using the expression of p-adic qintegral on Zp of these polynomials. 2. q-Bernoulli Numbers with Weight 0 and Bernstein Polynomials In the special case, α 0, the q-Bernoulli numbers with weight 0 will be denoted by β̃ 0 n,q β̃n,q. From 1.4 , 1.5 , and 1.6 , we note that ∞ ∑ n 0 β̃n,q t n! ∞ ∑ n 0 ∫


Introduction
Let p be a fixed prime number.Throughout this paper, Z p , Q p , and C p will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Q p , respectively.Let N be the set of natural numbers and Z N ∪ {0}.Let | • | p be a p-adic norm with |x| p p −r , where x p r s/t and p, s p, t s, t 1, r ∈ Q.In this paper, we assume that q ∈ C p with |1 − q| p < p −1/ p−1 so that q x exp x log q , and x q 1 − q x / 1 − q .Let UD Z p be the space of uniformly differentiable functions on Z p .For f ∈ UD Z p , the p-adic q-integral on Z p is defined by Kim as follows: 1 , we note that where f l df x /dx| x l , see 3, 6, 7 .In the special case, n 1, we get Throughout this paper, we assume that α ∈ Q.
The q-Bernoulli numbers with weight α are defined by Kim 8 as follows: with the usual convention about replacing β α q n with β α n,q .From 1.4 , we can derive the following equation: 1.5 By 1.1 , 1.4 , and 1.5 , we get The q-Bernoulli polynomials with weight α are defined by x y n q α dμ q y n l 0 n l q αlx x n−l q α β α l,q .1.7 By 1.6 and 1.7 , we easily see that Let C Z p be the set of continuous functions on Z p .For f ∈ C Z p , the p-adic analogue of Bernstein operator of order n for f is given by Abstract and Applied Analysis 3 where n, k ∈ Z see 1, 9, 10 .For n, k ∈ Z , the p-adic Bernstein polynomials of degree n are defined by B k,n x n k x k 1 − x n−k for x ∈ Z p , see 1, 10, 11 .In this paper, we consider Bernstein polynomials to express the p-adic q-integral on Z p and investigate some interesting identities of Bernstein polynomials associated with the q-Bernoulli numbers and polynomials with weight 0 by using the expression of p-adic qintegral on Z p of these polynomials.

q-Bernoulli Numbers with Weight 0 and Bernstein Polynomials
In the special case, α 0, the q-Bernoulli numbers with weight 0 will be denoted by β 0 n,q β n,q .From 1.4 , 1.5 , and 1.6 , we note that Z p e xt dμ q x q − 1 log q t log q qe t − 1 .

2.1
It is easy to show that t log q qe t − 1 where H n q −1 are the nth Frobenius-Euler numbers.By 2.1 and 2.2 , we get Therefore, we obtain the following theorem.
Theorem 2.1.For n ∈ Z , we have where H n q −1 are the nth Frobenius-Euler numbers.
From 1.5 , 1.6 , and 1.7 , we have with the usual convention about replacing β q n with β n,q .By 1.7 , the nth q-Bernoulli polynomials with weight 0 are given by From 2.6 , we can derive the following function equation: Thus, by 2.7 , we get that By the definition of p-adic q-integral on Z p , we see that x − 1 n dμ q x −1 n β n,q −1 .

2.9
Therefore, by 2.8 and 2.9 , we obtain the following theorem.

2.10
In particular, x −1, we get From 2.5 , we can derive the following equation:

2.12
Therefore, by 2.12 , we obtain the following theorem.
Theorem 2.3.For n ∈ N with n > 1, we have Taking the p-adic q-integral on Z p for one Bernstein polynomials in 1.9 , we get

2.14
From the symmetry of Bernstein polynomials, we note that

2.15
Let n > k 1.Then, by Theorem 2.3 and 2.15 , we get

2.16
By comparing the coefficients on the both sides of 2.14 and 2.16 , we obtain the following theorem.

2.17
In particular, when k 0, we have

2.18
Let m, n, k ∈ Z with m n > 2k 1.Then we see that

2.19
For m, n, k ∈ Z , we have

2.20
By comparing the coefficients on the both sides of 2.19 and 2.20 , we obtain the following theorem.−1 l β l,q 1 n m q − 1 log q − q q 2 β n m,q −1 .

2.21
In particular, when k / 0, we have By the same method above, we get

2.23
From the binomial theorem, we note that By comparing the coefficients on the both sides of 2.23 and 2.24 , we obtain the following theorem.Theorem 2.6.For s ∈ N, let k, n 1 , . . ., n s ∈ Z with n 1 n 2 • • • n s > sk 1.Then, we have

2.25
In particular, when k / 0, we have