On Aik–Summability

Abstract

In this paper, we introduce and investigate the concept of AIK-summability as an extension of AI∗-summability which was recently (2021) introduced by O.H.H.~Edely, where A=(ank)∞n,k=1 is a non-negative regular matrix and I and K represent two non-trivial admissible ideals in N. We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that AK-summability always implies AIK-summability whereas AI-summability not necessarily implies AIK-summability. Finally, we give a condition namely AP(I,K) (which is a natural generalization of the condition AP) under which AI-summability implies AIK-summability.