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 $B_n^{(r)}$ 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 $B_n^{(r)}∕(r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right))$, 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 $E_k^{(r)}(z)$ whose constant terms are the numbers $B_n^{(r)}∕(r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right))$. 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.