Hensel's lemma and the divisibility by primes of Stirling-like numbers
type: Item
instance of: scholarly article
title: Hensel's lemma and the divisibility by primes of Stirling-like numbers
zbMATH Open document ID: 0839.11061
author: Francis Clarke
published in: Journal of Number Theory
publication date: 1996-06-19T00:00:00Z
review text: Stirling-like number means \(T_p (n,k) = \sum (-1)^{k - j} {k \choose j} j^n\) summing with \(j\) from 0 to \(k\) and \(p \nmid j\), which \(p\)-adically approximates to \(k!S (n,k) = \sum (-1)^{k - j} {k \choose j} j^n\) with \(j\) from 0 to \(k\), where \(S(n,k)\) are the Stirling numbers. The author obtains results on the divisibility of \(T_p (n,k)\) and \(S(n,k)\) by primes \(p\) and gives detailed calculations for \(p = 2\) with \(k \leq 7\), and for \(p = 3\) with \(k \leq 5\), which generalise results of \textit{D. M. Davis} about exponent of 2 in \(T_2 (n,5)\) and \(T_2 (n,6)\) in [Proc. Am. Math. Soc. 110, No. 3, 597-600 (1990; Zbl 0712.11016)]. The essential idea is the observation that Davis' technique is in fact an application of Hensel's lemma. So he extends Hensel's lemma to analytic functions of a \(p\)-adic variable, and extracts an analytic function from \(T_p (n,k)\). He also conjectures that \(v_p (T_p (n,k)) = v_p (k!S (n,k))\) for all \(n \geq k\).
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 11B73
zbMATH DE Number: 765162
zbMATH Keywords: \(p\)-adic number
zbMATH Keywords: \(p\)-adic analytic theory
zbMATH Keywords: analytic functions of a \(p\)-adic variable
zbMATH Keywords: Stirling numbers
zbMATH Keywords: divisibility
zbMATH Keywords: Hensel's lemma
reviewed by: Xianke Zhang
MaRDI profile type: Publication
full work available at URL: jnth.1995.1056
OpenAlex ID: W2035632264
Wikidata QID: Q124816371
DOI: 10.1006/JNTH.1995.1056
description: scientific article; zbMATH DE number 765162