Misha Guysinsky, Thesis Supervisor Sergei Tabachnikov, Thesis Honors Advisor
Keywords:
bernoulli polynomial riemann zeta function
Abstract:
In this paper, we derived Bernoulli polynomials from differential operators e^{xd/dx}, which was originally considered for summation of powers. We proved that Bernoulli polynomials are the unique polynomials satisfying certain properties. This further enables
us to obtain a structure theorem for summing powers of integers. As an unusual case, we proved the Riemann zeta function could be represented in terms of Bernoulli numbers. Moreover, it has an analytic continuation by applying properties of Bernoulli polynomials.
