On the recursive sequence \( {x}_{n+1}=\frac{x_{n-\left(4k+3\right)}}{1+\prod_{t=0}^2{x}_{n-\left(k+1\right)t-k}} \)

Abstract

The solution of the difference equation

\( {x}_{n+1}=\frac{x_{n-\left(4k+3\right)}}{1+\prod_{t=0}^2{x}_{n-\left(k+1\right)t-k}},\kern0.5em n=0,1,2,\dots, \)

where x_−(4k+3), x_−(4k+2), . . . , x₋₁, x₀ ∈ (0, ∞) and k = 0, 1, . . . , is studied.