Hensel's lemma and the divisibility by primes of Stirling-like numbers
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 \(n \geq k\).
analytic functions of a \(p\)-adic variable
scientific article; zbMATH DE number 765162