ON <TEX>${mathcal{I}}$</TEX>-LACUNARY ARITHMETIC STATISTICAL CONVERGENCE

In this paper, we introduce arithmetic <TEX>${mathcal{I}}$</TEX>-statistically convergent sequence space <TEX>$A{mathcal{I}}SC$</TEX>, <TEX>${mathcal{I}}$</TEX>-lacunary arithmetic statistically convergent sequence space <TEX>$A{mathcal{I}}SC_{ heta}$</TEX>, strongly <TEX>${mathcal{I}}$</TEX>-lacunary arithmetic convergent sequence space <TEX>$AN_{ heta}[{mathcal{I}}]$</TEX> and prove some inclusion relations between these spaces. Futhermore, we give <TEX>${mathcal{I}}$</TEX>-lacunary arithmetic statistical continuity. Finally, we define <TEX>${mathcal{I}}$</TEX>-Ces&#x00E0;ro arithmetic summability, strongly <TEX>${mathcal{I}}$</TEX>-Ces&#x00E0;ro arithmetic summability. Also, we investigate the relationship between the concepts of strongly <TEX>${mathcal{I}}$</TEX>-Ces&#x00E0;ro arithmetic summability, strongly <TEX>${mathcal{I}}$</TEX>-lacunary arithmetic summability and arithmetic <TEX>${mathcal{I}}$</TEX> -statistically convergence.