On Generalized Difference Lacunary Statistical Convergence

A lacunary sequence is an increasing integer sequence θ = (kr) such that k0 =0, kr−kr−1 →∞ as r →∞. A sequence x is called Sθ(∆)− convergent to L provided that for each ε > 0, limr(kr − kr−1) {the number of kr−1 < k ≤ kr : |∆xk−L| ≥ ε} = 0, where ∆xk = ∆xk− ∆xk+1. The purpose of this paper is to introduce the concept of ∆ − lacunary statistical convergence and ∆-lacunary strongly convergence and examine some properties of these sequence spaces. We establish some connections between ∆-lacunary strongly convergence and ∆-lacunary statistical convergence. It is shown that if a sequence is ∆-lacunary strongly convergent then it is ∆-lacunary statistically convergent. We also show that the space Sθ(∆ ) may be represented as a [f, p, θ](∆) space.