On the dynamics of {\(x_{n+1}=\frac{a+x_{n-1}x_{n-k}}{x_{n-1}+x_{n-k}}\)} (Q2927961)

The author investigates the behavior of the solutions of the difference equationNEWLINENEWLINENEWLINE\[NEWLINE x_{n+1}=\frac{a+x_{n-1}x_{n-2}}{x_{n-1}+x_{n-k}},\quad n=0,1,2,\dots, NEWLINE\]NEWLINE where \(k\in \{1,2\}\), \(a\geq 0\), and \(x_{-i}>0\), \(j=0,1,\dots,k\). Furthermore, he distinguishes between the two cases \(a=0\) and \(a>0\).NEWLINENEWLINEThe results obtained are valid. However, if \(a>0\), then the substitution \(x_n=\sqrt{a}/z_n\) transforms the above equation into NEWLINE\[NEWLINE z_{n+1}=\frac{z_{n-1}+z_{n-k}}{1+z_{n-1}z_{n-k}},\quad n=0,1,2,\dots, NEWLINE\]NEWLINE which, if \(k\neq 1\), is a special case of the work done by \textit{K. S. Berenhaut} et al. [Appl. Math. Lett. 20, No. 1, 54--58 (2007; Zbl 1131.39006)], where a more general equation was investigated: NEWLINE\[NEWLINE y_n=\frac{y_{n-k}+y_{n-m}}{1+y_{n-k}y_{n-m}},\quad n=0,1,2,\dots. NEWLINE\]NEWLINE On the other hand, if \(a=0\), then the substitution \(x_n=1/z_n\) transforms the above equation into NEWLINE\[NEWLINE z_{n+1}= z_{n-1}+z_{n-2},\quad n=0,1,2,\dots. NEWLINE\]NEWLINE The latter equation is a special case of the the following linear equation NEWLINE\[NEWLINE y_{n+1}= y_{n-k}+y_{n-m},\quad n=0,1,2,\dots, NEWLINE\]NEWLINE where \(0\leq k\leq m\) and the initial conditions are positive. Furthermore, since \(y_{n+1}>y_{n-k}\) for all \(n\) and the sequence \(\{y_n\}\) is bounded away from \(0\), \(y_n \rightarrow \infty \) as \(n\rightarrow \infty\). Hence, \(x_n \rightarrow 0\) as \(n\rightarrow \infty\).