On generalizations of certain summability methods using ideals

(<i>f</i>, <i>p</i>)-Asymptotically Lacunary Equivalent Sequences with Respect to the Ideal <i>I</i>

“…Later it was further studied by Dass et al [15], Dems [16], Savas and Gumus [17], Kumar and Sharma [18], Kumar and Mursaleen [19] and many others.…”

𝐈𝟐 − 𝐒𝐓𝐀𝐓𝐈𝐒𝐓𝐈𝐂𝐀𝐋 And 𝐈𝟐 −Lacunary Statistical Convergence for Double Sequence of Order �

“…The class of all −lacunary statistically convergent sequences of order will be denoted by ( ) . Remark 2.3: For = 1 the definition coincides with −lacunary statistical convergence (see Das et al [10]). Theorem 2.1 (Das and Savas [11]): Let 0 < ≤ ≤ 1.…”

“…Remark 2.2: For = , ( ) −convergence coincides with statistical convergence of order (see Colak [6]). For an arbitrary ideal and for = 1 it coincides with −statistical convergence (see Das et al [10]). When = and = 1 it reduces to statistical convergence.…”

Asymptotically I-Lacunary statistical equivalent of order $\alpha$ for sequences of sets

“…More investigations in this direction and more applications of ideals can be found in [4,5,[16][17][18][19][20] where many important references can be found.…”

“…where |A| denotes the cardinality of A ⊂ N. Recently in ( [5] and [19]), we used ideals to introduce the concepts of I-statistical convergence and Ilacunary statistical convergence which naturally extend the notions of the above mentioned convergence. On the other hand, in [2] a different direction was given to the study of statistical convergence where the notion of statistical convergence of order α, 0 < α 1 was introduced by replacing n by n α in the denominator in the definition of statistical convergence.…”