A New Extension of the Sun-Zagier Result Involving Bell Numbers and Derangement Numbers

摘要

Let p be any prime and let a and n be positive integers with pn. We show that where B0, B1, ... are the Bell numbers and D0, D1,... are the derangement numbers. This extends a result of Sun and Zagier published in 2011. Furthermore, we prove that where Bk(x) = ∑l=0k S(k, l)xl is the Bell polynomial of degree k with S(k, l) (0 ≤l ≤ k) the Stirling numbers of the second kind, and Zp is the ring of all p-adic integers.