A note on the Bernoulli polynomials with the variable [x

Abstract

In this paper, we introduce the Bernoulli polynomials with the variable [x]q and we get polynomials related to the Bernoulli polynomials with the variable [x]q and the Bernoulli numbers as coefficients. By the p-adic integral on Zp, we define Bernoulli polynomials and we also get the some interesting properties. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of the Bernoulli polynomials Bn,q(x) in complex plane.