ON GENERALIZED DIFFERENCE LACUNARY STATISTICAL CONVERGENCE

Abstract

A lacunary sequence is an increasing integer sequence 0 = (k(r)) such that k(0) = 0, k(r)-k(r-1) -> infinity as r -> infinity. A sequence x is called S-theta (Delta(m))-convergent to L provided that for each epsilon > 0, lim(r) (k(r) - k(r-1))(-1) {the number of k(r-1) < k <= k(r) : vertical bar Delta(m) x(k)-L vertical bar >=epsilon} = 0, where Delta(m) x(k) = Delta(m-1) x(k) - m(-1) x(k+1). The purpose of this paper is to introduce the concept of Delta(m)-lacunary statistical convergence and Delta(m)-lacunary strongly convergence and examine some properties of these sequence spaces. We establish some connections between Delta(m) -lacunary strongly convergence and Delta(m)-lacunary statistical convergence. It is shown that if a sequence is Delta(m)-lacunary strongly convergent then it is Delta(m)-lacunary statistically convergent. We also show that the space S-theta (Delta(m)) may be represented as a [f, p, theta](Delta(m)) space.