Some New Formulae for Genocchi Numbers and Polynomials Involving Bernoulli and Euler Polynomials

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