A Linear Functional Equation of Third Order Associated with the Fibonacci Numbers

Abstract

Given a vector space $X$, we investigate the solutions $f:\mathbb{R}\to X$ of the linear functional equation of third order $f\left(x\right)=pf\left(x-1\right)+qf\left(x-2\right)+rf(x-3)$, which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.