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NEW RESULTS ON BERNOULLI NUMBERS OF HIGHER ORDER

Let r be any positive integer. In this paper, we revisite explicit and recurrence formulas satisfied by the Bernoulli numbers B n of higher order r. By using the unsigned Stirling numbers, we give them new forms and a clearer aspect. The second part of this study, consists in providing and proving analogues Kummer congruences and a von Staudt-Clausen theorem for the numbers B (r) n r(nr) , and investigating their numerators. Moreover, we study the p-integrality of these numbers. In the third part of this paper, we construct a new familly of Eisenstein series E k (z) whose constant terms are the numbers B (r) n r(nr) . 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 opens the way to several applications in algebraic number theory.