Lacunary Statistical Convergence of Order $$\alpha $$ for Generalized Difference Sequences and Summability Through Modulus Function

Abstract

In this paper, we define the space \(S_\theta ^\alpha (\Delta _v^m)\) of all \(\Delta _v^m\)-lacunary statistical convergent sequences of order \(\alpha \) and the space \(N_\theta ^\alpha (\Delta _v^m,p)\) of all strongly \(N_\theta (\Delta _v^m,p)\)-summable sequences of order \(\alpha \), where p is a positive real number. Some inclusion relations between these spaces have been obtained. We have studied the space \(\omega _\theta ^\alpha (\Delta _v^m,f,p)\) of all strongly \(\omega _\theta (\Delta _v^m,f,p)\)-summable sequences of order \(\alpha \) by using modulus function f and bounded sequence \((p_k)\) of positive real numbers with \(\displaystyle \inf _k p_k>0\). The inclusion relations between spaces \(\omega _\theta ^\alpha (\Delta _v^m,f,p)\) and \(S_\theta ^\alpha (\Delta _v^m)\) are also obtained.