Relation ships Classical, Hypergeometric and Fractional Bernoulli Numbers and Polynomials

Abstract

Bernoulli polynomials are named after the Swiss mathematician Jacob Bernoulli (1654- 1705). These are the class of polynomials Bn(x) defined by zexz e z − 1 = X∞ n=0 Bn(x) z n n! , for |z| < 2π. With Bn = Bn(0), the rational numbers Bn are called Bernoulli numbers. In 2008, Abdulkadir Hassen and Hiue D. Nguyen considered a generalization of Bn(x) called hy pergeometric Bernoulli polynomials of order N, Bn(N, x), defined by z N e xz/N! e z − TN−1(z) = X∞ n=0 Bn(N, x) z n n! . where TN (z) = X N k=0 z k k! . When N = 2, we obtain the class of polynomials Bn(2, x) first considered by F. T. Howard. In this thesis, we investigate the relation ships and differences classical, hypergeometric and fractional Bernoulli numbers, polynomials and their functional equations. We discuss several properties of these numbers and polynomials.