The p-adic analysis of Stirling numbers via higher order Bernoulli numbers

Generalized higher order Stirling numbers

p-Adic Measures and Bernoulli Numbers

The p-Stirling Numbers

The purpose of this article is to introduce p-Stirling numbers of the first and second kinds.

The 2-adic Valuation of Stirling Numbers

We analyze properties of the 2-adic valuations of S(n, k), the Stirling numbers of the second kind. A conjecture that describes patterns of these valuations for fixed k and n modulo powers of 2 is presented. The conjecture is established for k = 5.

Carlitz q-Bernoulli Numbers and q-Stirling Numbers

a+ dpZp = {x ∈ X | x ≡ a (mod dp N )}, where a ∈ Z lies in 0 ≤ a < dp , see [1-21]. The p-adic absolute value in Cp is normalized so that |p|p = 1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ Cp, then we assume |q − 1|p < p − 1 p−1 , so that q = exp(x log q) for |x|p ≤ 1. We use the notation [x]q = [x :...