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
