q-BERNOULLI NUMBERS AND POLYNOMIALS ASSOCIATED WITH GAUSSIAN BINOMIAL COEFFICIENT

Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ C, then we always assume |q| < 1. If q ∈ Cp, we normally assume |1− q|p < p − 1 p−1 , which implies that q = exp(x log q) for |x|p ≤ 1. Here, | · |p is the p-adic absolute value in Cp with |p|p = 1 p . The q-basic natural number are defined by [n]q = 1−q 1−q = 1 + q + · · · + q , ( n ∈ N), and q-factorial are also defined as [n]q! = [n]q ·[n−1]q · · · [2]q ·[1]q. In this paper we use the notation of Gaussian binomial coefficient as follows: