On Necessary and Sufficient Conditions for Absolute Matrix Summability

Abstract

This study gets a new general theorem related to necessary and sufficient conditions for $\varphi-{\mid D,\beta;\delta\mid}_{k}$ summability of the series $\sum a_n \lambda_n$ whenever the series $\sum a_n$ is summable $\varphi-{\mid C,\beta;\delta\mid}$, where $C=(c_{nv})$ and $D=(d_{nv})$ are two positive normal matrices, $k\geq 1$, $\delta\geq 0$ and $-\beta(\delta{k}+k-1)+k > 0$.

A Note on Summability of Infinite Series

“…Let us mention about some of the recent ones. Karakaş [5,6] , Kartal [7,8], Özarslan [9][10][11][12], Özarslan and Kartal [13,14], Özarslan, Şakar and Kartal [15] proved theorems on the sufficient conditions to absolute matrix summability of infinite series by using different class of sequences. The object of this paper is to get the necessary and sufficient conditions to absolute matrix summability.…”