On p -adic Integral Representation of q -Bernoulli Numbers Arising from Two Variable q -Bernstein Polynomials

: The q -Bernoulli numbers and polynomials can be given by Witt’s type formulas as p -adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q -Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q -Bernstein polynomials and show that they can be expressed in terms of the q -Bernoulli numbers and some special values of q -Bernoulli polynomials. This is generalized to the problem of evaluating any ﬁnite product of two variable q -Bernstein polynomials. Furthermore, some identities for q -Bernoulli numbers are found.


Introduction
Let p be a fixed prime number. Throughout this paper, N, Z p , Q p , and C p will denote the set of natural numbers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Q p , respectively. The p-adic norm | · | p is normalized as |p| p = 1 p . Assume that q is an indeterminate in C p such that |1 − q| p < p It is known that the q-number is defined by see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Please note that lim q→1 [x] q = x. 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 see [9,10].
From (3) and (10), we have with the usual convention about replacing B n q by B n,q . By (10), we easily get see [9]. As is known, the p-adic q-Bernstein operator is given by where n, k ∈ N ∪ {0}, x ∈ Z p , and f is a continuous function on Z p (see [7]). Here are called the p-adic q-Bernstein polynomials of degree n (see [7]). Please note that lim q→1 B k,n (x|q) = B k,n (x), where B k,n are the Bernstein polynomials (see [1,2,[18][19][20]22]).
Here we cannot go without mentioning that Phillips (see [16]) introduced earlier in 1997 a different version of q-Bernstein polynomials from Kim's. Let f be a function defined on [0, 1], q any positive real number, and let Then Phillips' q-Bernstein polynomial of order n for f is given by Many results of Phillips' q-Bernstein polynomials for q > 1 were obtained for instance in [14,15], while those for q ∈ (0, 1) were derived for example in [12,13]. However, all of these and other related papers deal only with analytic properties of those q-Bernstein polynomials and some applications of them.
The Volkenborn integral and the fermionic p-adic, the p-adic q-invariant and the fermionic p-adic q-invariant integrals introduced by Kim have been studied for more than twenty years. Numberous results of arithmetic or combinatorial nature have been found by Kim and his colleagues around the world.
The present and related paper (see [5,6]) concern about Kim's q-Bernstein polynomials which have some merits over Phillips'. Indeed, by considering p-adic integrals on Z p of them we can easily derive integral representations of q-Bernoulli numbers in the present paper, those of a q-analogue of Euler numbers in [5] and those of q-Euler numbers in [6]. These approaches also yield some identities for q-Bernoulli numbers, q-analogue of Euler numbers and q-Euler numbers. In conclusion, the Phillips' q-Bernstein polynomials are more analytic nature, while the Kim's are more arithmetic and combinatorial nature.
In this paper, we will study q-Bernoulli numbers and polynomials, which is introduced as p-adic invariant integrals on Z p , and investigate some properties for these numbers and polynomials. Also, we will consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we will consider p-adic integrals on Z p of any finite product of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. Furthermore, some identities for q-Bernoulli numbers will be found.

Some Integral Representations of q-Bernoulli Numbers and Polynomials
First, we consider the two variable q-Bernstein operator of order n which is given by where n ∈ N, and x 1 , Here, for n, k ≥ 0, are called two variable q-Bernstein polynomials of degree n (see [6,7]). In particular, this implies that B k,n (x 1 , where k ∈ N ∪ {0} (see [6,7]). From (13), we easily get For 1 ≤ k ≤ n − 1, we have the following properties (see [6,7]): From (13) and q-Bernstein operator, we note that where n ∈ N and f is a continuous function on Z p .
To see this, we first observe that Then (19) can be obtained as follows: It is easy to show that where j ∈ N ∪ {0} and x 1 , x 2 ∈ Z p . Indeed, by making use of (18), we see that From (2), we have By (10) and (21), we get Again, from (11) and (12), we can derive the following equation.
Thus, by (23), we obtain the following lemma.
By (2), (10) and (22), we get For n ∈ N with n > 1, by (21), Lemma 1, and (24), we have Let us take the double p-adic integral on Z p for the two variable q-Bernstein polynomials. Then we have Therefore, we obtain the following theorem.
Therefore, by Theorems 1 and 2, we obtain the following corollary.

Corollary 1.
For k ∈ N with k > 1, we have For m, n ∈ N ∪ {0}, we have Thus, by (29), we get Hence, we have the following proposition.
Let m, n, k ∈ N ∪ {0}. Then we get Thus, from (30), we have By (31), we have the following proposition.

Corollary 2.
For k ∈ N, we have For m ∈ N, let n 1 , n 2 , · · · , n m , k ∈ N ∪ {0}. Then we note that Thus, by (32), we have Therefore we obtain the following theorem.
Theorem 3. For n 1 , n 2 , · · · , n m ∈ N ∪ {0}, and k, m ∈ N with mk > 1, we have On the other hand, we easily get Therefore, by Theorem 3 and (33), we obtain the following corollary.
Corollary 3. For m, k ∈ N with mk > 1, we have

Conclusions
Here we studied q-Bernoulli numbers and polynomials which are different from the classical Carlitz q-Bernoulli numbers β n,q and polynomials β n,q (x), and arise naturally from some p-adic invariant integrals on Z p , as was shown in (10). After investigating some of their properties, we turned our attention to two variable q-Bernstein polynomials and operators, which was introduced by Kim and generalizes the single variable q-Bernstein polynomials and operators in [6]. As a preparation, we derived several properties of these polynomials and operators.
Next, we considered the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and showed that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This was further generalized to the problem of evaluating the product of two and that of an arbitrary number of two variable q-Bernstein polynomials. It was shown again that they can be expressed in terms of the q-Bernoulli numbers. Also, some identities for q-Bernoulli numbers were found along the way.
Finally, we would like to mention that, along the same line, in [5] we studied some properties of a q-analogue of Euler numbers and polynomials arising from the p-adic fermionic integrals on Z p . Then we considered p-adic fermionic integrals on Z p of the two variable q-Bernstein polynomials and of products of the two variable q-Bernstein polynomials, and showed that they can be expressed in terms of the q-analogues of Euler numbers.
Author Contributions: T.K. and D.S.K. conceived the framework and structured the whole paper; T.K. wrote the paper; C.S.R. and Y.Y. checked the results of the paper; D.S.K., C.S.R., Y.Y. and T.K. completed the revision of the article.