The 2-adic analysis of Stirling numbers of the second kind via higher order Bernoulli numbers and polynomials

Abstract

Several new estimates for the [Formula: see text]-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of [Formula: see text] in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.

New results on the p-adic valuation of Stirling numbers

“…In recent years several papers appeared in which the p-adic properties of these numbers and the 2-adic properties, in particular have been investigated. Various efforts were made to determine congruences and the p-adic valuations of both kinds of Stirling numbers, e.g., [4], [5], [6], [7], [9], [10], [13], [15], [16], [17], [18], [21], [24], [25], [26], and [12], [14], [19], [20], [23], etc. Some of these results are universal and deal with ν p (S(n, k)) and ν p (s(n, k)) with no particular restrictions on n and k while other results focus on cases when n is in the form of cp h with c, h ∈ Z + .…”

“…The major development came in a sequence of papers by Adelberg (cf. [4] and [5]) that depended on his previous work on higher order Bernoulli numbers and polynomials, and on the relation between higher order Bernoulli numbers and Stirling polynomials. In [4, Theorem 2.2], Adelberg generalized (1.1) for arbitrary primes and replaced the term 2 h with the condition that S(n, k) is a so-called minimum zero case (MZC).…”

“…A remarkable short and instructive proof of Theorem 1.1 is presented as a special case in [4,Theorem 2.2]. To deal with the case of c > 1 and odd, Adelberg also introduced the notion of almost minimum zero case (AMZC) in [5] that resulted in a short proof of Theorem 1.2. We define these concepts in relation with inequality (1.3).…”

“…Although (1.3) is general in n and k, it was Adelberg who developed a general theory in [4] and [5] that removed the limitations due to the assumed nature of n = c2 h and k : 1 ≤ k ≤ 2 h in S(n, k) that were used in Theorems 1.1 and 1.2.…”

“…He generalized it to odd primes, and proved a criterion for when it is sharp, i.e., when S(n, k) is a minimum zero case (MZC). He also added an additional term to improve the estimate in all cases where it is not sharp, namely he proved in [5,Theorem 3.3] that…”