Bernoulli Polynomials and Ramanujan Summation

Abstract

The concept of Bernoulli numbers has appeared in several counting problems in mathematics. Bernoulli numbers are vastly used in many mathematical problems, especially dealing with Riemann zeta function, as Leonhard Euler in the eighteenth century, proved that the zeta function values for positive even integers are connected to even subscript Bernoulli numbers. As a natural generalization of Bernoulli numbers, we discuss Bernoulli polynomials. Bernoulli numbers occur as constant terms of the respective Bernoulli polynomials. This paper introduces the concept of Bernoulli numbers and Bernoulli polynomials and discusses few properties concerning them. In Sect. 2.1, the relation between successive Bernoulli numbers is obtained. This will help us in generating successive Bernoulli numbers from the initial Bernoulli number \(B_{0} = 1\). Using these Bernoulli numbers, we obtain a nice formula for generating successive Bernoulli polynomials in Sect. 3.1. In Sect. 4.1, the concept that the Bernoulli numbers are obtained by corresponding polynomials of Bernoulli at t = 1 has been established. Finally, in Sect. 5.1, the connection between Riemann zeta function and Bernoulli polynomials has been proved. Further, five figures are included to justify the truth of Sect. 4.1. The main focus of this paper is to establish the relation between Bernoulli polynomials and Ramanujan Summation.