Construction and Characterization of Oscillatory Chain Sequences

TL;DR

This paper explores a new class of oscillatory chain sequences near 1/4, proving fixed point existence, convergence, and divergence properties, expanding the theoretical understanding of such sequences.

Contribution

It generalizes Szwarc's framework to oscillatory sequences around 1/4, providing necessary and sufficient conditions and explicit solutions.

Findings

Proves existence of a fixed point for the critical map.

Establishes convergence properties linking oscillations to parameters.

Constructs sequences with divergent series outside previous bounds.

Abstract

This paper initiates a theoretical investigation of $\frac{1}{4}$-oscillatory chain sequences $\{a_n\}$, generalizing Szwarc's classical framework for non-oscillatory chains \cite{Sz94, Sz98, Sz02, Sz03} to sequences fluctuating around $\frac{1}{4}$. We prove the existence of a fixed point for the critical map $f(x)=1-\frac{1}{4x}$ and establish convergence properties linking oscillatory behavior to parameter sequences $\{g_n\}$. A complete characterization is provided via a necessary and sufficient condition, exemplified by explicit solutions $a_n=\frac{1}{4}\left(1+(-1)^{n}\varepsilon_{n}\right)$. Crucially, we construct oscillatory chain sequences for which the series $\sum_{n=1}^{\infty} \left(a_n - \frac{1}{4}\right)$ diverges, demonstrating fundamentally different behavior outside the hypothesis $a_n \ge \frac{1}{4}$ required by Chihara's bound.