An Application of Absolute Matrix Summability to Infinite Series

Source abstract

This paper investigates sufficient conditions for absolute matrix summability factors of an infinite series. Starting from a known theorem on absolute Riesz summability factors, the study extends the result to the more general $\varphi$-|B; $\delta$|k summability framework, where B = ($b,\eta,\omega$) is a positive normal matrix. The development uses an almost increasing sequence ($X\eta$) together with conditions imposed on the matrix entries, the factor sequence ($\xi\eta$), the sequence ($Z\eta$) of (C, 1) means associated with ($\eta{d\eta}$), and the parameters k and $\delta$. Two lower semimatrices generated by B are introduced to express the relevant sequence-to-sequence and series-to-series transformations. Under the stated hypotheses, the main theorem establishes that the factored series $\sum{d_\eta\xi_\eta}$ is $\varphi$-|B; $\delta$|k summable for k ≥ 1 and 0 ≤ $\delta$ < 1/k. The proof applies Abel’s transformation and decomposes the transformed expression into four components. Each component is then estimated using the matrix assumptions, properties of almost increasing sequences, and Holder’s inequality. The resulting bounds establish the required convergence without introducing assumptions beyond those stated in the theorem. The argument therefore connects the matrix conditions directly with the summability conclusion. Appropriate choices of $\delta$, $\varphi$$\eta$, and the matrix entries reduce the theorem to previously known absolute matrix and absolute Riesz summability results, showing that the established result provides a common framework for these special cases.