Some relations for universal Bernoulli polynomials

Abstract

In this paper, we consider a generalization of the Bernoulli polynomials, which we call universal Bernoulli polynomials. They are related to the Lazard universal formal group. The corresponding numbers by construction coincide with the universal Bernoulli numbers. They turn out to have an important role in complex cobordism theory. They also obey generalizations of the celebrated Kummer and Clausen–von Staudt congruences. We derive a formula on the integral of products of higher-order universal Bernoulli polynomials. As an application of this formula, the Laplace transform of periodic universal Bernoulli polynomials is presented. Moreover, we obtain the Fourier series expansion of higher-order universal Bernoulli function.