On some properties of the series ∑k=0∞ knxk and the Stirling numbers of the second kind Original Research Article

We partially characterize the rational numbers x and integers n ⩾ 0 for which the sum ∑k=0∞ knxk assumes integers. We prove that if ∑k=0∞ knxk is an integer for x = 1 − a/b with a, b > 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain νa(n−k)!⩾νa(n!)−k+1 for all integers k, 2 ⩽ k ⩽ n, and a ⩾ 3, provided a is odd or divisible by 4, where va(m) denotes the exponent of the highest power of a which divides m, for m and a > 1 integers. New identities are also derived for the Stirling numbers, e.g., we show that ∑k=02nk! S(2n, k) −12k=0, for all integers n ⩾ 1.