Some Identities on the Twisted h , q-Genocchi Numbers and Polynomials Associated with q-Bernstein Polynomials

Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of padic rational integer, Qp denotes the ring of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N {0}. Let Cpn {ζ | ζpn 1} be the cyclic group of order p and let


Introduction
Let p be a fixed odd prime number.Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Z p denotes the ring of padic rational integer, Q p denotes the ring of p-adic rational numbers, and C p denotes the completion of algebraic closure of Q p , respectively.Let N be the set of natural numbers and Z N {0}.Let C p n {ζ | ζ p n 1} be the cyclic group of order p n and let The p-adic absolute value is defined by |x| 1/p r , where x p r s/t r ∈ Q and s, t ∈ Z with s, t p, s p, t 1 .In this paper we assume that q ∈ C p with |q − 1| p < 1 as an indeterminate.

International Journal of Mathematics and Mathematical Sciences
The q-number is defined by Note that lim q → 1 x q x.Let UD Z p be the space of uniformly differentiable function on Z p .For f ∈ UD Z p , Kim defined the fermionic p-adic q-integral on Z p as follows: , where f n x f x n for n ∈ N.For k, n ∈ Z and x ∈ 0, 1 , Kim defined the q-Bernstein polynomials of the degree n as follows: see 13-15 .For h ∈ Z and ζ ∈ T p , let us consider the twisted h, q -Genocchi polynomials as follows: Then, G h n,q,ζ x is called nth twisted h, q -Genocchi polynomials.In the special case, x 0 and G h n,q,ζ 0 G h n,q,ζ are called the nth twisted h, q -Genocchi numbers.
In this paper, we give the fermionic p-adic integral representation of q-Bernstein polynomial, which are defined by Kim 13 , associated with twisted h, q -Genocchi numbers and polynomials.And we construct some interesting properties of q-Bernstein polynomials associated with twisted h, q -Genocchi numbers and polynomials.

On the Twisted h, q -Genocchi Numbers and Polynomials
From 1.6 , we note that

2.1
We also have Therefore, we obtain the following theorem.

2.4
Therefore, we obtain the following theorem.

Theorem 2.2. For n ∈ Z and ζ ∈ T p , one has
From 1.5 , one gets the following recurrence formula:

2.6
Therefore, we obtain the following theorem.
with usual convention about replacing From Theorem 2.3, we note that

2.8
Therefore, we obtain the following theorem.
Theorem 2.4.For n ∈ Z and ζ ∈ T p , one has 2.9 Remark 2.5.We note that Theorem 2.4 also can be proved by using fermionic integral equation 1.4 in case of n 2.
By 2.4 and Theorem 2.2, we get

2.10
Therefore, we obtain the following theorem.
Theorem 2.6.For n ∈ Z and ζ ∈ T p , one has

2.11
Let n ∈ N. By Theorems 2.4 and 2.6, we get Therefore, we obtain the following corollary.
Corollary 2.7.For n ∈ Z and ζ ∈ T p , one has By 1.5 , we get the symmetry of q-Bernstein polynomials as follows: (see [11]).

International Journal of Mathematics and Mathematical Sciences
Thus, by Corollary 2.7 and 2.14 , we get

2.15
From 2.15 , we have the following theorem.
Theorem 2.8.For n ∈ Z and ζ ∈ T p , one has

2.16
For n, k ∈ Z with n > k, fermionic p-adic invariant integral for multiplication of two q-Bernstein polynomials on Z p can be given by the following:

2.17
From Theorem 2.8 and 2.17 , we have the following corollary.
Corollary 2.9.For n ∈ Z and ζ ∈ T p , one has

2.19
From 2.19 , we have the following theorem.
Theorem 2.10.For n ∈ Z and ζ ∈ T p , one has

2.20
Let n 1 , n 2 , k ∈ Z with n 1 n 2 > 2k, fermionic p-adic invariant integral for multiplication of two q-Bernstein polynomials on Z p can be given by the following: 1 .

2.21
From Theorem 2.10 and 2.21 , we have the following corollary.

International Journal of Mathematics and Mathematical Sciences
Corollary 2.11.For n 1 , n 2 , k ∈ Z and n 1 n 2 > 2k, one has 2.22