On a new closed formula for the solution of second order linear difference equations and applications

TL;DR

This paper introduces a new closed-form solution for second-order linear difference equations using matrix theory, extending formulas like Binet's to negative indices and deriving new summation identities.

Contribution

It presents a novel closed formula for second-order difference equations, extending classical results like Binet's formula to negative indices and providing new summation formulas.

Findings

New closed formula for second-order difference equations

Extension of Binet's formula to negative integers

Derivation of new summation identities

Abstract

In this note, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of this type such as the Fibonacci and Lucas sequences. As applications; we show that Binet's formula, in this case, is valid for negative integers as well. Finally, we find new summation formulas relating the elements of such sequences.