Congruence relation between Stirling numbers of the first and second kinds

Abstract

This paper consists of certain congruence properties of Stirling numbers of the first and second kinds. Some congruence relations between s(n, k) and S(n, k) for different modulo are obtained through their generating functions. We also present some exact p-adic valuations of s(n, k) and S(n, k) for some cases, mainly when \(n-k\) is divisible by \(p-1\) for odd prime p. Some estimates of the p-adic valuation of these two numbers are also presented when \(p-1\) does not divide \(n-k\).