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THE INTEGRALITY OF THE VALUES OF BERNOULLI POLYNOMIALS AND OF GENERALISED BERNOULLI NUMBERS

Published online by Cambridge University Press:  01 January 1997

FRANCIS CLARKE  and
I. SH. SLAVUTSKII
FRANCIS CLARKE
Affiliation:
Department of Mathematics, University of Wales, Singleton Park, Swansea SA2 8PP
I. SH. SLAVUTSKII
Affiliation:
Centre for Absorption in Science, Jerusalem, Israel
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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.

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 29 , Issue 1 , January 1997 , pp. 22 - 24
Copyright
© The London Mathematical Society 1997

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