On a rational recursive sequence with parameter near the boundary

This note studies existence of positive prime periodic solutions of higher order for rational recursive equations of the form y_n = A + y_n−k/y_n−m, n = 0,1,2,…, with y_−s, y_−s+1,…, y₋₁ ∈ (0, ∞), k odd and m ∈ {1,2,3,4,…}, where s = max {k,m}. In particular, we show that for k ≥ 5, odd, m ≥ 1, gcd (k,m) = 1 and sufficiently small A > 0, there exist periodic solutions with prime period 2m* + U_m*, for some m*, where U_m = min {i ∈ ℕ: i (i + 1) ≥ 2m}. A value of m* > (k − 1)²/2 + m is given explicitly.