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The 2-adic analysis of Stirling numbers of the second kind via higher order Bernoulli numbers and polynomials
International Journal of Number Theory ( IF 0.7 ) Pub Date : 2020-07-18 , DOI: 10.1142/s1793042121500044
Arnold Adelberg 1
Affiliation  

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.

中文翻译:

通过高阶伯努利数和多项式对第二类斯特林数的二次分析

证明了对第二类斯特林数的 [公式:见正文]-adic 估值的几个新估计。这些估计,连同它们何时尖锐的标准,导致了几个已知定理及其证明的改进,以及新的定理,包括 Lengyel 长期存在的公开猜想。估计和标准都取决于我们之前对高阶伯努利多项式系数分母中[公式:见文本]的幂的分析。也证明了第一类斯特林数的相应估计。对渐近案例给予了一些关注,这将在后续出版物中进一步探讨。
更新日期:2020-07-18
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