Nontrivial solutions of a higher-order rational difference equation

AbstractWe prove that, for every k ∈ ℕ, the following generalization of the Putnam difference equation $$ x_{n + 1} = \frac{{x_n + x_{n - 1} + \cdots + x_{n - (k - 1)} + x_{n - k} x_{n - (k + 1)} }} {{x_n x_{n - 1} + x_{n - 2} + \cdots + x_{n - (k + 1)} }}, n \in \mathbb{N}_0 , $$ has a positive solution with the following asymptotics $$ x_n = 1 + (k + 1)e^{ - \lambda ^n } + (k + 1)e^{ - c\lambda ^n } + o(e^{ - c\lambda ^n } ) $$ for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λk+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.