JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 24, No. 4, December 2011 SOME PROPERTIES INVOLVING THE HIGHER ORDER q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α, β) VIA THE p-ADIC q-INTEGRAL ON Z p Jong Jin Seo* and Serkan Aracı** Abstract. The main properties of this paper is to describe the higher order q-Genocchi polynomials with weight (α, β). However, we derive some interesting properties concerning this type of poly- nomials. 1. Introduction, definitions and notations The main motivations of this paper are the papers [8], [14] by Kim et al. and Hwang et al., in which they introduced and studied on higher order q-Euler numbers and polynomials with weight α and higher order q-Bernoulli numbers and polynomials with weight α, respectively. By using q-Volkenborn integral, Kim introduced the q-Bernoulli num- bers and polynomials with weight α which are derived some interest- ing properties of q-Bernoulli numbers and polynomials with weight α. However, several mathematicians have studied on the special numbers and poynomials with weight α(see for details [3],[4], [8], [9], [10], [11], [14],[17], [22]). Assume that p is a fixed odd prime number. Throughout this paper Z p , Q p , C and C p , will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Q p . Let N be the set of natural numbers and Z + = N ∪{0}. Let v p be the normalized exponential valuation of C p with |p| p = p -vp(p) = 1 p (see [1-6,8-17]). If q ∈ C, then Received October 15, 2011; Accepted November 24, 2011. 2010 Mathematics Subject Classification: Primary 05A10, 11B65, 28B99, 11B68, 11B73. Key words and phrases: Genocchi numbers and polynomials, q-Genocchi numbers and polynomials, higher order q-Genocchi numbers and polynomials.