We determine the best Ulam constant for the higher order linear difference equation \(x_{n+p}=a_1x_{n+p-1}+\cdots +a_px_n+b_n\) in a Banach space, with scalar (real or complex) constant coefficients \(a_1,\ldots , a_p\) (p is a fixed positive integer), in the case when all its characteristic complex roots are outside the unit circle. In this way we improve the known Ulam stability results for this equation and obtain quite precise measure of the difference between approximate solutions of a discrete dynamical system employing the equation (i.e., solutions of a bounded perturbation of the system) and the exact solutions to it (i.e., solutions of the system without perturbation). Moreover, as an auxiliary result we give an explicit expression of the general solution of the equation, and at the end of the paper we formulate an open problem.
