The p-adic Analysis of Stirling Numbers via Higher Order Bernoulli\n Numbers

2018/05/02 by Arnold Adelberg, Adelberg, Arnold
Mathematics · #11A07 #11B73 (Primary) 11B68 #11S05 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1805.00995

openalex publication_date 2018/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we use our previous study of the higher order Bernoulli\nnumbers B_n^(l) to investigate the p-adic properties of the Stirling\nnumbers of the second kind S(n,k). For example, we give a new, greatly\nsimplified proof of the formula \ν₂(S(2^h,k))=d₂(k)-1 if 1\≤ k \≤ 2^h,\nand generalize this result to arbitrary primes p. We also consider the\nStirling numbers of the first kind s(n,k), with new results analogous to\nthose for the Stirling numbers of the second kind. New mod p congruences for\nStirling numbers of both kinds are also given.\n

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