On the difference equation \(x_{n} = x_{n - 2}/(b_{n} + c_{n}x_{n - 1}x_{n - 2})\) (Q427019)

The author studies the difference equation NEWLINE\[NEWLINEx_n= x_{n-2}(b_n+ c_n x_{n-1} x_{n-2})^{-1},\quad n\in\mathbb{N}_0,NEWLINE\]NEWLINE where \((b_n)_n\) and \((c_n)_n\) are sequence periodic with period two and the initial values \(x_{-2}\), \(x_{-1}\) are real numbers. If \(c_n= 0\), \(n\in\mathbb{N}_0\), \(b_n\neq 0\), \(n\in\mathbb{N}_0\), he obtains NEWLINE\[NEWLINE\begin{aligned} x_n= b^{-1}_n x_{n-2},\quad & n\in\mathbb{N}_0,\\ x_{2m}= x_{-2} b^{-m-1}_0,\quad & x_{2m+1}= x_{-1} b^{-m-1}_1,\quad m\in\mathbb{N}_0,\end{aligned}NEWLINE\]NEWLINE and he studies the cases when \(x_n\to 0\), \(x_{2m}\to 0\), \(x_{2m+ 1}\to 0\), \(x_{2m}\to+\infty\), \(x_{2m+1}\to+\infty\) if \((x_{2m})_m\) is constant, \((x_{2m+1})_m\) is constant, \((x_{2m})_m\) is two-periodic, \((x_{2m+1})_m\) is two-periodic and other cases. If \(b_n= 0\), \(n\in\mathbb{N}_0\), \(c_n\neq 0\), \(n\in\mathbb{N}_0\), he obtains NEWLINE\[NEWLINEx_n= x_{n-2}(c_n x_{n-1} x_{n-2})^{-1},\qquad n\in\mathbb{N}_0,NEWLINE\]NEWLINE and similar results. The author studies the case \(b_n\neq 0\) and \(c_n\neq 0\), \(n\in\mathbb{N}\), and uses this case in applications: \(b_0 b_1= -1\), \(b_0 b_1= 1\), \(b_0 b_1\neq \pm 1\).