Some New Results on Absolute Summability Factors

Abstract

In this paper, we establish the general summability factor theorems related to generalized absoluteCesàro summability |𝐶, 𝛼, 𝛽|𝑘 and absolute factorable matrix summability |𝐴𝑓,𝜑𝑛|𝑘methods for 𝑘 ≥ 1,𝛼 + 𝛽 > −1, where (𝜑𝑛) is arbitrary sequence of positive real constants and 𝐴𝑓 = (𝑎𝑛𝑣) is a factorablematrix such that 𝑎𝑛𝑣 = 𝑎̂𝑛𝑎𝑣for 0 ≤ 𝑣 ≤ 𝑛, 𝑎𝑛𝑣 = 0 for 𝑣 > 𝑛, (𝑎̂𝑛) and (𝑎𝑛) are any sequences ofreal numbers. Also, absolute factorable summability method includes all absolute Riesz summabilityand absolute weighted summability methods in the special cases. Therefore, not only some well knownresults but also several new results for absolute Cesàro and weighted means are obtained as corollaries.