Real Zeros of Bernoulli Polynomials on the Closed Interval [0,1]

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.