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The Integrality of the Values of Bernoulli Polynomials and of Generalised Bernoulli Numbers
Author(s) -
Clarke Francis,
Slavutskii I. SH.
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609396001695
Subject(s) - mathematics , bernoulli's principle , bernoulli polynomials , bernoulli process , bernoulli number , pure mathematics , combinatorics , difference polynomials , orthogonal polynomials , physics , thermodynamics
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.