The elementary symmetric functions of a reciprocal polynomial sequence

摘要

Erd?s 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 ≥ 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 ≥ 2 being an integer and n = 1.