On the asymptotics of the difference equation yn(1 + yn-1... yn-k+1)= yn-k

Abstract

We show that the difference equation where k ∈ ℕ\{1}, has a positive solution converging to zero, by finding a finite asymptotic expansion of the solution. We also show that if p0 = 0 and p1,..., pk-1 are arbitrarily given positive numbers, then there exists a solution of the equation such that the subsequences (ykm+i)m∈ℕ0; i = 0,..., k - 1, have partial sums of exponential power series as finite asymptotic expansions. Finally, we sketch the case when q of the limits pi:= lim m→∞ykm+i (q > 1) are vanishing, where we determine only heuristically the next term of the asymptotic expansion.