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In this paper, we will deal with some new formulae for two product Genocchi polynomials together with both Euler polynomials and Bernoulli polynomials. We get some applications for Genocchi polynomials. Our applications possess a number of interesting properties to study in Theory of Analytic numbers which we express in the present paper.

Abstract:
We consider the weighted -Genocchi numbers and polynomials. From the construction of the weighted -Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials. 1. Introduction Let be a fixed odd prime number. Throughout this paper, , , , and , will, respectively, denote the ring of -adic integers, the field, of -adic rational numbers, the complex number field and the completion of algebraic closure of . Let be the normalized exponential valuation of such that (see [1–16]). As well-known definition, the Euler numbers and Genocchi numbers are defined by with the usual convention of replacing by and with the usual convention of replacing by . We assume that with and that the -number of is defined by (see [1–19]). In [9], Kim introduced ordinary fermionic -adic integral on , and he studied some interesting relations and identities related to -extension of Euler numbers and polynomials. In [8], he also introduced the -extension of the ordinary fermionic -adic integral on and he investigated many physical properties related to -Euler numbers and polynomials. Recently, Kim firstly introduced the meaning of the weighted -Euler numbers and polynomials associated with the weighted -Bernstein polynomials by using the fermionic invariant -adic integral on (see [14, 15]). In [16], Ryoo tried to study the weighted -Euler number and polynomials by the same method of Kim et al. in [14] and the -extension of the fermionic -adic invariant integrals on . As well-known properties, the Genocchi numbers are integers. The first few Genocchi numbers for are . The first few prime Genocchi numbers are ？3 and 17, which occur for and 8. There are no others with . These properties are very important to study in the area of fermionic distribution and -adic numbers theory. By this reason, many mathematicians and physicians have studied Genocchi and Euler numbers which are in the different areas. By the same motivation, we consider weighted -Genocchi polynomials and numbers by using the fermionic -adic -integral on which are constructed by Kim and Ryoo (cf. [8, 16]). In this paper, we consider the -Genocchi numbers and polynomials with weighted . From the construction of the weighted -Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials. 2. The Weighted -Genocchi Numbers and Polynomials Let be the space of uniformly differentiable functions and, for , the fermionic -adic invariant integral of on is defined by Kim as follows:

Abstract:
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

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We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials.

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Recently mathematicians have studied some interesting relations between -Genocchi numbers, -Euler numbers, polynomials, Bernstein polynomials, and -Bernstein polynomials. In this paper, we give some interesting identities of the twisted -Genocchi numbers, polynomials, and -Bernstein polynomials with weighted .

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We give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation. 1. Introduction The history of Genocchi numbers can be traced back to Italian mathematician Angelo Genocchi (1817–1889). From Genocchi to the present time, Genocchi numbers have been extensively studied in many different context in such branches of Mathematics as, for instance, elementary number theory, complex analytic number theory, homotopy theory (stable homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), -adic analytic number theory ( -adic -functions), and quantum physics (quantum groups). The works of Genocchi numbers and their combinatorial relations have received much attention [1–11]. For showing the value of this type of numbers and polynomials, we list some of their applications. In the complex plane, the Genocchi numbers, named after Angelo Genocchi, are a sequence of integers that are defined by the exponential generating function: with the usual convention about replacing by , is used. When we multiply with in the left-hand side of (1), then we have where are called Genocchi polynomials. It follows from (2) that , , , , , , , , and for (for details, see [7–9]). Differentiating both sides of (1), with respect to , then we have the following: On account of (1) and (3), we can easily derive the following: By (1), we get Thanks to (4) and (5), we acquire the following equation (6): It is not difficult to see that By expression of (7), then we have (see [1–25]). Let be the -dimensional vector space over . Probably, is the most natural basis for . From this, we note that is also good basis for space . In [14], Kim et al. introduced the following integrals: where and are called Bernoulli polynomials and Euler polynomials, respectively. Also, they are defined by the following generating series: with and , symbolically. By (10), then we have Here and are called Bernoulli numbers and Euler numbers, respectively. Additionally, the Bernoulli and Euler numbers and polynomials have the following identities: (for details, see [6, 11, 13–15, 17, 19]). By (11), we have the following recurrence relations of Euler and Bernoulli numbers, as follows: where is the Kronecker’s symbol defined by In the complex plane, we can write the following: By (15), we have by comparing coefficients on the both sides of the above equality, then we have (see [6]). Via (17), our

Abstract:
We give some new identities for (h; q)-Genocchi numbers and polynomials by means of the fermionic p-adic q-integral on Zp and the weighted q-Bernstein polynomials.

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Recently many mathematicians are working on Genocchi polynomials andGenocchi numbers. We define a new type of twisted q-Genocchi numbers and polynomialswith weight and weak weight and give some interesting relations of the twistedq-Genocchi numbers and polynomials with weight and weak weight . Finally, we findrelations between twisted q-Genocchi zeta function and twisted Hurwitz q-Genocchi zetafunction.