On Log-Concave Operator Acting on Sequences and Series

In this paper, we investigate the properties of sequences and series under the action of the log-concave operator $\mathcal{L}$. We explore the relationship between the convergence of a sequence $(a_k)$ and the convergence of sequences and series derived by applying $\mathcal{L}$ iteratively. These results demonstrate the strong regularity properties of log-concave sequences and provide a framework for analyzing the convergence of sequences and series derived from the log-concave operator. The findings have implications for combinatorics, probability, optimization, and related fields, opening new avenues for further research on the behavior of log-concave sequences and their associated operators.