Abstract
Modifying the definition of density functions is one method used to generalise statistical convergence. In the present study, we use sequences of modulus functions and order $\alpha \in \left( 0,1\right] $ to introduce a new density. Based on this density framework, we define strong $(f_k)$-lacunary summability of order $\alpha $ and $(f_k)$-lacunary statistical convergence of order $\alpha $ for a sequence of modulus functions $(f_k)$. This concept holds an intermediate position between the usual convergence and the statistical convergence for lacunary sequences. We also establish inclusion theorems and relations between these two concepts in the study.
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