On the recursive sequence \(x_{n+1}=\frac{A}{\prod^ k_{i=0}x_{n-i}}+\frac{1}{\prod^{2(k+1)}_{j=k+2}x_{n-j}}\).
For the difference equation in the title with \(A>1\) it is shown that every positive solution converges to a \((k+2)\)-periodic solution. By means of the substitution \(y_n= x_nx_{n-1}\cdots x_{n-k-1}\) this implies that every positive solution of \(y_{n+1}= A+ y_n/y_{n-k-1}\) converges to \(A+1\).
scientific article; zbMATH DE number 2076839