Applications of fourier series and zeta functions to Genocchi polynomials

Abstract

In this paper, we firstly consider the properties of Genocchi polynomials, Fourier series and Zeta functions. In the special cases, we see that the Fourier series yield Zeta functions. From here, we show that zeta functions for some special values can be computed by Genocchi polynomials. Secondly, we consider the Fourier series of periodic Genocchi functions. For odd indexes of Genocchi functions, we construct good links between Genocchi functions and Zeta function. Finally, since Genocchi functions reduce to Genocchi polynomials over the interval [0,1), we see that Zeta functions have integral representations in terms of Genocchi polynomials.