The 2-Adic Analysis of Stirling Numbers of the Second Kind via Higher Order Bernoulli Numbers and polynomials

TL;DR

This paper develops new 2-adic valuation estimates for Stirling numbers of both kinds using higher order Bernoulli numbers, leading to improved theorems and new results in number theory.

Contribution

It introduces novel 2-adic valuation estimates for Stirling numbers based on higher order Bernoulli polynomials, enhancing existing theorems and establishing new ones.

Findings

New bounds for 2-adic valuations of Stirling numbers of the second kind

Criteria for sharpness of the estimates

Extension of results to Stirling numbers of the first kind

Abstract

Several new estimates for the 2-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. The estimates and criteria all depend on our previous analysis of powers of 2 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.