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

We give some interesting identities on the twisted ( )-Genocchi numbers and polynomials associated with -Bernstein polynomials. 1. Introduction Let be a fixed odd prime number. Throughout this paper, we always make use of the following notations: denotes the ring of rational integers, denotes the ring of -adic rational integer, denotes the ring of -adic rational numbers, and denotes the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the cyclic group of order and let The -adic absolute value is defined by , where ( and with ( ) = ( ) = ( ) = 1). In this paper we assume that with as an indeterminate. The -number is defined by (see [1–15]). Note that . Let be the space of uniformly differentiable function on . For , Kim defined the fermionic -adic -integral on as follows: (see [2–6, 8–15]). From (1.3), we note that (see [4–6, 8–12]), where for . For and , Kim defined the -Bernstein polynomials of the degree as follows: (see [13–15]). For and , let us consider the twisted ( )-Genocchi polynomials as follows: Then, is called th twisted ( )-Genocchi polynomials. In the special case, and are called the th twisted ( )-Genocchi numbers. In this paper, we give the fermionic -adic integral representation of -Bernstein polynomial, which are defined by Kim [13], associated with twisted ( )-Genocchi numbers and polynomials. And we construct some interesting properties of -Bernstein polynomials associated with twisted ( )-Genocchi numbers and polynomials. 2. On the Twisted -Genocchi Numbers and Polynomials From (1.6), we note that We also have Therefore, we obtain the following theorem. Theorem 2.1. For and , one has with usual convention about replacing by . By (1.6) and (2.1) one gets Therefore, we obtain the following theorem. Theorem 2.2. For and , one has From (1.5), one gets the following recurrence formula: Therefore, we obtain the following theorem. Theorem 2.3. For and , one has with usual convention about replacing by . From Theorem 2.3, we note that Therefore, we obtain the following theorem. Theorem 2.4. For and , one has Remark 2.5. We note that Theorem 2.4 also can be proved by using fermionic integral equation (1.4) in case of . By (2.4) and Theorem 2.2, we get Therefore, we obtain the following theorem. Theorem 2.6. For and , one has Let . By Theorems 2.4 and 2.6, we get Therefore, we obtain the following corollary. Corollary 2.7. For and , one has By (1.5), we get the symmetry of -Bernstein polynomials as follows: (see [11]). Thus, by Corollary 2.7 and (2.14), we get From (2.15), we have the following