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The Integrality of the Values of Bernoulli Polynomials and of Generalised Bernoulli Numbers

The Integrality of the Values of Bernoulli Polynomials and of Generalised Bernoulli Numbers Abstract In [1] Almkvist and Meurman proved a result on the values of the Bernoulli polynomials (Theorem 5 below). Subsequently, Sury [5] and Bartz and Rutkowski [2] have given simpler proofs. In this paper we show how this theorem can be obtained from classical results on the arithmetic of the Bernoulli numbers. The other ingredient is the remark that a polynomial with rational coefficients which is integer-valued on the integers is Z(p)-valued on Z(p). Here Z(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. An application is given in Section 3 to the arithmetic of generalised Bernoulli numbers. 1991 Mathematics Subject Classification 11B68. © London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

The Integrality of the Values of Bernoulli Polynomials and of Generalised Bernoulli Numbers

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References (5)

Publisher
Oxford University Press
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609396001695
Publisher site
See Article on Publisher Site

Abstract

Abstract In [1] Almkvist and Meurman proved a result on the values of the Bernoulli polynomials (Theorem 5 below). Subsequently, Sury [5] and Bartz and Rutkowski [2] have given simpler proofs. In this paper we show how this theorem can be obtained from classical results on the arithmetic of the Bernoulli numbers. The other ingredient is the remark that a polynomial with rational coefficients which is integer-valued on the integers is Z(p)-valued on Z(p). Here Z(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. An application is given in Section 3 to the arithmetic of generalised Bernoulli numbers. 1991 Mathematics Subject Classification 11B68. © London Mathematical Society

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Jan 1, 1997

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