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February 2022 New results on Bernoulli numbers of higher order
Chouaib Khattou, Abdelmejid Bayad, Mohand Ouamar Hernane
Rocky Mountain J. Math. 52(1): 153-170 (February 2022). DOI: 10.1216/rmj.2022.52.153


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.


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Chouaib Khattou. Abdelmejid Bayad. Mohand Ouamar Hernane. "New results on Bernoulli numbers of higher order." Rocky Mountain J. Math. 52 (1) 153 - 170, February 2022.


Received: 25 January 2021; Revised: 29 June 2021; Accepted: 5 July 2021; Published: February 2022
First available in Project Euclid: 19 April 2022

Digital Object Identifier: 10.1216/rmj.2022.52.153

Primary: 11B68 , 11B73 , 11M36

Keywords: Bernoulli number of higher order , Clausen and von Staudt congruence , Eisenstein series , Kummer congruence , numerator of Bernoulli number of higher order , p-integrality , Stirling number

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.52 • No. 1 • February 2022
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