Tauberian theorems for the (A)(C, β) summability methods

In this study, we investigate two Tauberian theorems that extend the classical Tauberian theorems for the Abel summability method. The aim of these theorems is to derive the (C, α) summability of a sequence from the product of its Abel and Cesàro summability methods of order β, where β>α>-1 and α,β∈R. In the special case where β>α=0, a condition is provided for the transition from (A)(C, β) summability to convergence. These results not only generalize existing Tauberian theorems but also offer new insights into the interplay between Abel and Cesàro summability methods. Furthermore, the theorems contribute to the broader understanding of summability techniques in sequence analysis and their applications in mathematical series.