Asian Journal of Mathematics and Computer Research

This paper establishes a general summability factor theorem for the ϕ − |B; δ|_k summability of an infinite series associated with a positive normal matrix B. The study considers the sequence of partial sums of an infinite series and the corresponding matrix transformation determined by B. A generalised framework is developed for sequences (ϕ_n), (p_n), (λ_n), and (\(\gamma\)_n) under prescribed boundedness, monotonicity, and growth conditions. The main theorem replaces the earlier weighted mean setting with a broader positive normal matrix setting and assumes that ϕ_np_n ≍ P_n, together with additional conditions on the associated lower semimatrices \(\bar{B}\) and \(\hat{B}\). The proof is based on Abel’s transformation and estimates obtained through Holder’s inequality, after decomposing the transformed difference into four components. Each component is shown to satisfy the required convergence condition under hypotheses (12)–(18) and the assumptions inherited from the corresponding weighted mean theorem. The result demonstrates that the series \(\sum\) a_nλ_n(\(\gamma\)n)−1 is ϕ − |B; δ|_k summable for k ≥ 1 and 0 ≤ δ < 1/k. For δ = 0 with the specified choices of ϕn and bnv, the theorem yields a weighted mean matrix analogue of the known result. The study is confined to positive normal matrices satisfying the stated structural conditions, leaving wider matrix classes for further investigation.