A Study of Second-Order Linear Recurrence Sequences via Continuants

TL;DR

This paper reinterprets second-order linear recurrence sequences as continuants from continued fractions, deriving their generating functions, Binet formulas, and identities, thus providing a new perspective on Lucas sequences.

Contribution

It introduces a novel continuant-based framework for analyzing second-order linear recurrence sequences, including derivations of generating functions and identities.

Findings

Derived the generating function for continuants.

Established Binet formulas for these sequences.

Reformulated Lucas sequence identities using continuants.

Abstract

This paper presents a reinterpretation of a second-order linear recurrence sequence as a sequence of continuants derived from the convergents to a continued fraction. As a result, we are able to derive the generating function and Binet formula for continuants. Using this result, we provide a continuant-based formulation for well-known identities associated with Lucas sequences.