Nontrivial solutions of a higher-order rational difference equation

Abstract

We prove that, for every k ∈ ℕ, the following generalization of the Putnam difference equation xn+1 = xn + xn-1 +...+ xn-(k-1) + xn-kxn-(k+1)/x nxn-1 + xn-2 + ... + xn-(k+1) n ∈ ℕ0 has a positive solution with the following asymptotics xn = 1 + (k+1)e-cλn + o(e e -cλn) for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ k+2 - λ - 1 belonging to the interval (1, 2). Using this result, we prove that the equation has a positive solution which is not eventually equal to 1. Also, for the case k = 1, we find all positive eventually equal to unity solutions to the equation.