On the Recursive Sequence \( {x}_{n+1}=\frac{x_{n-\left(k+1\right)}}{1+{x}_n{x}_{n-1}\dots {x}_{n-k}} \)

Abstract

A solution of the following difference equation is investigated:

\( {x}_{n+1}=\frac{x_{n-\left(k+1\right)}}{1+{x}_n{x}_{n-1}\dots {x}_{n-k}},n=0,1,2,\dots \)

where x_−(k+1); x_−k; : : : ; x₋₁; x₀ ???? (0;∞) and k = 0; 1; 2; : : : .