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arXiv · 1805.00995

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

Arnold Adelberg
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Abstract

In this paper, we use our previous study of the higher order Bernoulli numbers $B_n^{(l)}$ to investigate the $p$-adic properties of the Stirling numbers of the second kind $S(n,k)$. For example, we give a new, greatly simplified proof of the formula $\nu_2(S(2^h,k))=d_2(k)-1$ if $1\le k \le 2^h$, and generalize this result to arbitrary primes $p$. We also consider the Stirling numbers of the first kind $s(n,k)$, with new results analogous to those for the Stirling numbers of the second kind. New mod $p$ congruences for Stirling numbers of both kinds are also given.

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Arnold Adelberg. 2018-05-02. The $p$-adic Analysis of Stirling Numbers via Higher Order Bernoulli Numbers. https://arxiv.org/abs/1805.00995

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