The Value of Bernoulli Polynomials at Rational Numbers

Abstract

We give a short elementary proof of a result of Almkvist and Meurman [1] on an integrality property of the values taken by the Bernoulli polynomials at a rational number. We use a lemma on the divisibility properties of certain binomial coefficients which seems to be of independent interest. 0. Introduction As is well known, the Bernoulli polynomials Bn(i) defined by Xe Z occur naturally while summing powers of the natural numbers. They also appear in other places, like in the evaluation of the Riemann zeta function at even integers, or while finding out whether or not a prime number is regular. As such, the properties of these polynomials are of some number-theoretic interest. Recently, G. Almkvist and A. Meurman proved the following result in [1]. THEOREM. Writing Bn(t) = BJJ) Bn(0), we have for all h,k,ne N, The purpose of this note is to give a short and completely elementary proof of this theorem. REMARK. AS observed in [1], it is enough to prove the theorem with the assumption that h = 1, since we have the addition formula Bn(x+y)= m 0 From now on, we write an = k Bn I -1 for simplicity. We employ two different \kj recursions for the numbers an and a lemma on the divisibility properties of certain binomial coefficients which seems to be of independent interest. Received 10 February 1992; revised 7 April 1992. 1991 Mathematics Subject Classification 11B68. Bull. London Math. Soc. 25 (1993) 327-329