A note on | A | k summability factors of infinite series

A weighted mean matrix, denoted by (N , pn), is a lower triangular matrix with entries pk/Pn, where {pk} is a nonnegative sequence with p0 > 0, and Pn := ∑n k=0 pk. Mishra and Srivastava [1] obtained sufficient conditions on a sequence {pk} and a sequence {λn} for the series ∑ anPnλn/npn to be absolutely summable by the weighted mean matrix (N , pn). Bor [2] extended this result to absolute summability of order k ≥ 1. Unfortunately, an incorrect definition of absolute summability was used. In this note, we establish the corresponding result for a nonnegative triangle, using the correct definition of absolute summability of order k ≥ 1, (see [3]). As a corollary, we obtain the corrected version of Bor’s result. Let A be an infinite lower triangular matrix. We may associate with A two lower triangular matrices A and Â, whose entries are defined by