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 : q] = 1− q 1− q . For f ∈ C(Zp) = {f | f ′ ∈ C(Zp)}, let us start with the expressions 1 [pN ]q ∑