Abstract
In this paper, we defined the concepts of lacunary $\mathcal{I}^{\ast}$-convergence and strongly lacunary $\mathcal{I}^{\ast}$-convergence. We investigated the relations between strongly lacunary $\mathcal{I}$-convergence and strongly lacunary $\mathcal{I}^{\ast}$-convergence. Also, we defined the concept of strongly lacunary $\mathcal{I}^{\ast}$-Cauchy sequence and investigated the relations between strongly lacunary $\mathcal{I}$-Cauchy sequence and strongly lacunary $\mathcal{I}^{\ast}$-Cauchy sequence.
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