arXiv:0710.4976v1 [math.NT] 26 Oct 2007 q-BERNOULLI NUMBERS AND POLYNOMIALS ASSOCIATED WITH GAUSSIAN BINOMIAL COEFFICIENT Taekyun Kim Abstract. The first purpose of this paper is to present a systemic study of some fami- lies of multiple q-Bernoulli numbers and polynomials by using multivariate q-Volkenborn integral (= p-adic q-integral) on Z p . From the studies of these q-Bernoulli numbers and polynomials of higher order we derive some interesting q-analogs of Stirling number identities. §1. Introduction Let q be regarded as either a complex number q ∈ C or a p-adic number q ∈ C p . If q ∈ C, then we always assume |q| < 1. If q ∈ C p , we normally assume |1 - q| p <p − 1 p-1 , which implies that q x = exp(x log q) for |x| p ≤ 1. Here, |·| p is the p-adic absolute value in C p with |p| p = 1 p . The q-basic natural number are defined by [n] q = 1−q n 1−q =1+ q + ··· + q n−1 , ( 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: (1)  n k  q = [n] q ! [n - k] q ![k] q ! = [n] q · [n - 1] q ··· [n - k + 1] q [k] q ! . Note that lim q→1 ( n k ) q = ( n k ) = n·(n−1)···(n−k+1) n! . The Gaussian coefficient satisfies the following recursion formula: (2)  n +1 k  q =  n k - 1  q + q k  n k  q = q n−k  n k - 1  q +  n k  q , cf. [1-23]. Key words and phrases. q-Bernoulli numbers, q-Volkenborn integrals, q-Euler numbers, q-Stirling numbers. 2000 AMS Subject Classification: 11B68, 11S80 This paper is supported by Jangjeon Research Institute for Mathematical Science(JRIMS-10R-2001) Typeset by A M S-T E X 1