Some Properties of Hypergeometric Bernoulli Numbers of Order 2

Abstract

The 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! , |z| < 2π where the variables z and x can be real or complex and the rational number Bn = Bn(0) are called Bernoulli numbers. For any positive integer N, a generalization of Bn(x) called hypergeometric Bernoulli poly nomials of order N, Bn(N, x), are defined by the generating functions as: z N e xz N!(e z − TN−1(z)) = X∞ n=0 Bn(N, x) z n n! where TN (z) = PN k=0 z k k! is the Taylor polynomial of order N of the exponential function e z . When N=2, we obtain the class of polynomials Bn(2, x) called hypergeometric Bernoulli polynomials of order 2. In this thesis, we mainly focus to this class of polynomials. We establish additional properties of hypergeometric Bernoulli numbers of order 2. We briefly explain the relation ship between Bernoulli numbers and hypergeometric Bernoulli numbers of order 2