The elementary symmetric functions of a reciprocal polynomial sequence

Abstract

Erdös and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m,1/(m+d),...,1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\ge 4$, then none of the elementary symmetric functions of $1/m,1/(m+d),...,1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1),1/f(2),...,1/f(n)$ is an integer except for $f(x)=x^m$ with $m\ge 2$ being an integer and $n=1$.

4 pages. To appear in Comptes Rendus Mathematique