Several Explicit Formulae of Bernoulli Polynomials

Abstract

The Bernoulli polynomials are named after the Swiss mathematician Jacob Bernoulli (1654 – 1705). These are the class of polynomials Bn(x) defined by text e t − 1 = X∞ n=0 Bn(x) t n n! , |t| < 2π, where the variables t and x are complex and the rational number Bn = Bn(0) are called Bernoulli numbers. This thesis provides an overview of Bernoulli numbers and Bernoulli polynomials. The purpose of this thesis to obtain possible extension formulae of Bernoulli polynomials depending on the value k which is greater or equal to n. In addition, the basic definitions about properties of Bernoulli numbers and Bernoulli polynomials are the part of this thesis. In this study we define some properties of Bernoulli numbers and Bernoulli polynomials and explicit of several formulae of Bernoulli polynomials. Finally, the main result we consider the proof of several explicit formulae of Bernoulli polynomials depending on the values of k and n (3.4) with different examples. That is; for 0 ≤ n ≤ k we have Bn(x) = 1 k + 1 X k i=0 X i j=0 (−1)i+j