On the periodicity of the solution of a rational difference equation

In this paper, some cases on the periodicity of the rational difference equation S_{n+1}=S_{n-p}(((aS_{n-q}+bS_{n-r}+cS_{n-s})/(dS_{n-q}+eS_{ n-r}+fS_{n-s}))), are investigated, where a, b, c, d, e, f ∈(0,∞). The initial conditions S_{-p}, S_{-p+1},...,S_{-q}, S_{-q+1},...,S_{-r}, S_{-r+1},...,S_{-s},...,S_{-s+1},...,S₋₁ and S₀ are arbitrary positive real numbers such that p>q>r>s≥0. Some numerical examples are provided to illustrate the theoretical discussion.