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On the rational recursive sequence \(x_{n+1}=\Big ( A+\sum _{i=0}^{k}\alpha _{i}x_{n-i}\Big ) \Big / \sum _{i=0}^{k}\beta _{i}x_{n-i}\). (English) Zbl 1199.39025
Summary: The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation \[ x_{n+1}=\bigg ( A+\sum _{i=0}^{k}\alpha _{i}x_{n-i}\bigg ) \Big / \sum _{i=0}^{k}\beta _{i}x_{n-i},\;\;n=0,1,2,\dots \] where the coefficients \(A\), \(\alpha _{i}\), \(\beta _{i}\) and the initial conditions \(x_{-k},x_{-k+1},\dots ,x_{-1},x_{0}\) are positive real numbers, while \(k\) is a positive integer number.

MSC:
39A22 Growth, boundedness, comparison of solutions to difference equations
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
39A30 Stability theory for difference equations
39A23 Periodic solutions of difference equations
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