# On the difference equation ${x}_{n+1}=\frac{{a}_{0}{x}_{n}+{a}_{1}{x}_{n-1}+\cdots +{a}_{k}{x}_{n-k}}{{b}_{0}{x}_{n}+{b}_{1}{x}_{n-1}+\cdots +{b}_{k}{x}_{n-k}}$

Mathematica Bohemica (2008)

• Volume: 133, Issue: 2, page 133-147
• ISSN: 0862-7959

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## Abstract

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In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence ${x}_{n+1}=\frac{{a}_{0}{x}_{n}+{a}_{1}{x}_{n-1}+\cdots +{a}_{k}{x}_{n-k}}{{b}_{0}{x}_{n}+{b}_{1}{x}_{n-1}+\cdots +{b}_{k}{x}_{n-k}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n=0,1,\cdots \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}$ where the parameters ${a}_{i}$ and ${b}_{i}$ for $i=0,1,\cdots ,k$ are positive real numbers and the initial conditions ${x}_{-k},{x}_{-k+1},\cdots ,{x}_{0}$ are arbitrary positive numbers.

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