### A note on generalized ${\left|A\right|}_{k}$-summability factors for infinite series.

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This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 $$\underset{r}{lim}\frac{1}{{h}_{r}}\left\{k\in {I}_{r}:\left|\frac{{x}_{{\sigma}^{k}\left(m\right)}}{{y}_{{\sigma}^{k}\left(m\right)}}-L\right|\u2a7e\in \right\}=0$$ uniformly in m = 1, 2, 3, ..., (denoted by x $$\stackrel{{S}_{\sigma ,\theta}}{\sim}$$ y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent...

In normed linear space settings, modifying the sequential definition of continuity of an operator by replacing the usual limit "$lim$" with arbitrary linear regular summability methods $\mathbf{G}$ we consider the notion of a generalized continuity ($({\mathbf{G}}_{\mathbf{1}},{\mathbf{G}}_{\mathbf{2}})$-continuity) and examine some of its consequences in respect of usual continuity and linearity of the operators between two normed linear spaces.

The main object of this paper is to introduce and study some sequence spaces which arise from the notation of generalized de la Vallée–Poussin means and the concept of a modulus function.

In this paper we introduce a new concept of $\lambda $-strong convergence with respect to an Orlicz function and examine some properties of the resulting sequence spaces. It is also shown that if a sequence is $\lambda $-strongly convergent with respect to an Orlicz function then it is $\lambda $-statistically convergent.

The purpose of this paper is to introduce some new generalized double difference sequence spaces using summability with respect to a two valued measure and an Orlicz function in $2$-normed spaces which have unique non-linear structure and to examine some of their properties. This approach has not been used in any context before.

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