Zayed, E. M. E.; El-Moneam, M. A. On the rational recursive sequence \( x_{n+1}=\dfrac {\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-k}} {\beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-k}} \). (English) Zbl 1224.39015 Math. Bohem. 135, No. 3, 319-336 (2010). Summary: The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation \[ x_{n+1}=\frac {\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-k}} {\beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-k}}, \quad n=0,1,2,\dots, \] where the coefficients \(\alpha _{i},\beta _{i}\in (0,\infty )\) for \( i=0,1,2,\) and \(l\), \(k\) are positive integers. The initial conditions \( x_{-k}, \dots , x_{-l}, \dots , x_{-1}, x_{0} \) are arbitrary positive real numbers such that \(l<k\). Some numerical experiments are presented. Cited in 4 Documents MSC: 39A20 Multiplicative and other generalized difference equations 39A22 Growth, boundedness, comparison of solutions to difference equations 39A23 Periodic solutions of difference equations 39A30 Stability theory for difference equations 65Q10 Numerical methods for difference equations Keywords:rational difference equation; boundedness; period two solution; convergence; global stability; positive solutions; numerical experiments PDFBibTeX XMLCite \textit{E. M. E. Zayed} and \textit{M. A. El-Moneam}, Math. Bohem. 135, No. 3, 319--336 (2010; Zbl 1224.39015) Full Text: EuDML EMIS