Monotonicity of solutions to second order linear difference equations with constant coefficients

TL;DR

The study addresses when monotone properties hold for solutions of $a_{n+2}-a a_{n+1}+b a_n=0$ with real parameters by deriving closed-form expressions in terms of the roots $\alpha_\pm$, analyzing asymptotics, and reframing monotonicity through a discrete Riccati equation. It provides necessary and sufficient conditions on $(a,b)$ and initial data for monotone relations such as $a_n\le a_{n+1}$ and related inequalities, including treatment of the double-root and complex-root scenarios. A key contribution is the Fibonacci-number characterization as a unique irreducible boundary case on the parameter domain, linking monotone properties to structural properties of the recurrence. These results offer a general framework for Fibonacci-like sequences and connect monotonicity criteria to number-theoretic concepts like Pisot numbers and irreducibility.

Abstract

We describe some monotone properties of solutions to second order linear difference equations with real constant coefficients. As an application, we give a characterization of the Fibonacci numbers.

Figures (8)

• Figure 1: $D_{1}$
• Figure 2: $D_{1}^{\prime }$
• Figure 3: $D_{2}$
• Figure 4: $D_{2}^{\prime }$
• Figure 5: $D_{3}$

Theorems & Definitions (33)

• Example 1.1
• Example 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof