Generalized lacunary statistical convergence of order β of difference sequences of fractional order

In this paper, using a modulus function we generalize the concepts of ∆m−lacunary statistical convergence and ∆m−lacunary strongly convergence (m ∈ N) to ∆α−lacunary statistical convergence of order β with the fractional order of α and ∆α−lacunary strongly convergence of order β with the fractional order of α ( where 0 < β ≤ 1 and α be a fractional order).