New results on Bernoulli numbers of higher order

Abstract

Let r be any positive integer. In this paper, we revisit explicit and recurrence formulas satisfied by the Bernoulli numbers Bn(r) of higher order r. By using the unsigned Stirling numbers, we give them new forms and a clearer description. The second part of this study consists of providing and proving analogues of Kummer congruences and a von Staudt–Clausen theorem for the numbers Bn(r)∕(rnr), as well as investigating their numerators. Moreover, we study the p-integrality of these numbers. In the third part of this paper, we construct a new family of Eisenstein series Ek(r)(z) whose constant terms are the numbers Bn(r)∕(rnr). We prove a congruence for these new Eisenstein series. This generalizes the classical von Staudt–Clausen’s and Kummer’s congruences of Eisenstein series. Our study leads to several applications in algebraic number theory.