New definitions about \(A^{\mathcal I}\)-statistical convergence with respect to a sequence of modulus functions and lacunary sequences (Q2305837)

Summary: In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of \(\mathcal I\)-statistical convergence, which is a recently introduced summability method. The names of our new methods are \(A^{\mathcal I}\)-lacunary statistical convergence and strongly \(A^{\mathcal I}\)-lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by \(S_\theta^A (\mathcal I,F)\) and \(N_\theta^A (\mathcal I,F)\), respectively. We give some inclusion relations between \(S^A (\mathcal I,F)\), \(S_\theta^A (\mathcal I,F)\) and \(N_\theta^A (\mathcal I,F)\). We also investigate Cesáro summability for \(A^{\mathcal I}\) and we obtain some basic results between \(A^{\mathcal I}\)-Cesáro summability, strongly \(A^{\mathcal I}\)-Cesáro summability and the spaces mentioned above.