Abstract:
Bernoulli numbers and Bernoulli polynomials are named after the Swiss
mathematician Jacob Bernoulli (1654-1705). He introduced these num bers and polynomials in his book Ars Conjectandi, published posthu mously (Basel, 1713). These class of polynomials Bn(x) can be defined
as follows;
wewx
e
w − 1
=
X
∞
n=0
Bn(x)
w
n
n!
for |w| < 2π,
where the variables x and w can be real or complex numbers. The
rational numbers Bn(0) = Bn are known as Bernoulli numbers. So for
any positive integer n, Bn(x) are called Bernoulli polynomials of degree
n. It has also a relation with Hurwitz’s zeta fuction,
ζ(1 − n, a) = −Bn(a)
n
for n > 1, 0 < a ≤ 1.
On the interval [0, 1], this relation can be further explained by Fourier
Series expression as follows;
Bn(a) = −n!
(2πi)
n
X
|k|≥1
e
2πika
k
n
=
−2n!
(2π)
n
X
∞
k=1
cos(2πka −
πs
2
)
k
n
for 0 ≤ a ≤ 1, n > 1.
In this thesis, we find real zeros of Bernoulli polynomials. Also, it will
be successfully implemented by using the properties of Bernoulli poly nomials. We relate the result with real zeros of Hurwitz’s zeta function
using trigonometric identities, in this case we use gragh to explain the
relations. The real zeros in the closed interval [0, 1] are presented using
tables of variation. Some numerical examples are given to demonstrate
the validity and applicability of the method.
