On Log-Concave Operator Acting on Sequences and Series
Piero Giacomelli
TL;DR
The paper studies how the log-concave operator $L$, defined by $L(a_k)=a_k^2-a_{k-1}a_{k+1}$, acts on sequences and their iterates, and how this affects convergence properties of both the sequences and their associated series. It develops a framework for $i$-fold and infinite log-concavity, proving that if $a_k\to L$, then $L^i(a_k)\to 0$ for all $i$, and that under log-concavity combined with convergence of $\sum a_k$, all $\sum L^i(a_k)$ converge with $L^i(a_k)$ decaying exponentially. The work also establishes that convergence properties of $a_k$ can imply convergence of derived sequences under certain monotonicity and boundedness hypotheses, and it proves preservation lemmas for $L$ that help extend these results to infinite log-concavity. Additionally, it analyzes the $L$-series and demonstrates conditions under which the series $\sum a_k$ and its $L$-derived series are simultaneously convergent, offering a toolset with potential applications in combinatorics, probability, and optimization. Open questions on necessary conditions and rate bounds point to fruitful directions for future research.
Abstract
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.