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More on a rational recurrence relation. (English) Zbl 1069.39024
The paper is mainly concerned with the long-term behavior of the solutions of the difference equation \[ x_{n+1}=\frac{x_{n-1}}{1+x_{n-1}x_n}, \quad n=0,1,2,\dots \tag{de} \]
where the initial conditions \(x_{-1},~x_0\) are real numbers. An explicit formula of the solution of equation (de) is given by
\[ x_n=\begin{cases} x_{-1} \frac{\prod_{i=0}^{[(n+1)/2]-1} (2ix_{-1}x_0+1)} {\prod_{i=0}^{[(n+1)/2]-1} ((2i+1)x_{-1}x_0+1)} &\quad\text{if } n\text{ is odd}\\ x_0 {\frac{\prod_{i=1}^{n/2} ((2i-1)x_{-1}x_0+1)} {\prod_{i=1}^{n/2} (2i x_{-1}x_0+1)}} &\quad\text{if } n\text{ is even} \end{cases} \tag{ef} \] Formula (ef) was, first, presented and justified by means of mathematical induction by C. Cinar [Appl. Math. Comput. 158, 809–812 (2004; Zbl 1066.39007)]. However, the author of this note derives the formula in a constructive way by introducing the change of variables \(x_{n+1}x_n=z_n\). Furthermore, by utilizing formula (ef), he shows that
1) If \(x_{-1}x_0 \neq 0,~ -1/n,~n\in \mathbb{N}\), then every solution of equation (de) converges to zero.
2) If \(x_{-1}x_0=0\), then every solution of equation (de) is periodic of period 2.
Finally, as noted by the author, the difference equation
\[ y_{n+1}=\frac{by_{n-1}}{b+cy_{n-1} y_n},\quad n=0,1,2, \] where \(b\) and \(c\) are positive real numbers, can be reduced to equation (de) by introducing the change of variables \(y_n=\sqrt{b/c}x_n\).

MSC:
39A20 Multiplicative and other generalized difference equations
39A11 Stability of difference equations (MSC2000)
Citations:
Zbl 1066.39007
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