On the families of numbers with respect to Orlicz functions

Abstract

Summable families of numbers were defined
by E. H. Moore \cite{P}, who also showed that an infinite series of real or
complex numbers converges unconditionally if and only if it is
summable. In this paper, we introduce an extension of power series methods in the sense of summable families. As applications, we construct the spaces of families of numbers with respect to Orlicz functions and study some expansions of $P$-strongly convergent and $P$-statistically convergent series with respect to Orlicz functions. Our results are natural extensions of the sequence spaces defined by Orlicz, which are introduced in \cite{PC} and \cite{Sa,Sa1}.