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

✍ Scribed by Tamás Lengyel

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 491 KB
• Volume: 150
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(95)00193-z

✦ Synopsis

We partially characterize the rational numbers x and integers n >/0 for which the sum ~--~o k"xk • go n assumes integers. We prove that if ~k=0 k x k 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 Stifling number of the second kind, S(n,k). More specifically, we obtain Va ((n --k)!S(n, n -k)) >1 va(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 Stifling numbers, e.g., we show that ~2k~ 0 k! S(2n, k) ( _ ½)k = 0, for all integers n >~ 1.