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

Abstract

Genocchi numbers Gn and Genocchi polynomials Gn(x) can be derived respectively by the following exponential generating functions. 2t et + 1 = 1X n =0 Gntn n!; 2tetx et + 1 = 1X n =0 Gn(x)tn n!; (jtj < π) In the usual notation, Bn(x) and En(x) are denoted by Bernoulli and Euler polynomials of degree n in x, defined by the generating functions respectively. tetx et − 1 = 1X n =0 Bn(x)tn n!; (jtj < 2π) 2etx et + 1 = 1X n =0 En(x)tn n!; (jtj < π) where Bn and En are respectively Bernoulli and Euler numbers. In 2014, Araci et al. [3] established the Basic properties of Bernoulli numbers and polynomials, Euler numbers and polynomials and Genocchi numbers and polynomials are discussed. In this thesis, we can derive some identities relationship between the Genocchi polynomials to Bernoulli and Euler polynomials and we get some applications for Genocchi polynomials.