On the solutions of second order difference equations with variable coefficients

TL;DR

This paper investigates solutions to second order linear difference equations with variable coefficients, providing closed form solutions via continued fractions and deriving new formulas for π².

Contribution

It introduces elementary methods to solve such difference equations and presents novel generalized continued fraction formulas for π².

Findings

Closed form solutions using finite continued fractions

Elementary proof based on quadratic shift operator factoring

New generalized continued fraction formulas for π²

Abstract

In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are elementary and based on factoring a quadratic shift operator. As an application, we obtain two new generalized continued fraction formulas for the mathematical constant $\pi^2$.