The study uses a summability method is utilized to characterize the set (|R, pn, δ|k, |R, qn, δ|k) for k ≥ 1 and any arbitrary positive sequences (pn). Advanced results have been derived from previously established findings. This is significant because absolute summability is crucial for assessing the Bounded Input Bounded Output (BIBO) stability of systems. By imposing specific conditions on the main result, sufficient criteria for the summability of |R, pn|k and |C, 1|k have been established, reinforcing the relevance and application of this study.