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On the asymptotics of the difference equation \(y_{n}(1 + y_{n - 1} \cdots y_{n - k + 1}) = y_{n - k}\). (English) Zbl 1220.39011

The existence of positive solutions of the difference equation
\[ y_{n}=\frac{y_{n-k}}{1+y_{n-1}\dots y_{n-k+1}}~,\quad n\in\mathbb N_{0},\tag{\(*\)} \]
where \(k\in\mathbb N\backslash \{1\}\), converging to zero is studied. The main result of this article is the following Theorem: For every \(k\in\mathbb N\backslash \{1\}\), equation (\(*\)) has a positive solution with the following asymptotics
\[ y_{n}=\left( \frac{k}{(k-1)n}\right) ^{1/(k-1)}\left( 1+a\frac{\ln n}{n}+b\frac{\ln ^{2}n}{n^{2}}+o\left(\frac{\ln ^{2}n}{n^{2}}\right)\right), \]
where for \(k=2p+1\),
\[ a=\frac{2p+1}{8p^{2}}\text{ and }b=\frac{(2p+1)^{3}}{128p^{4}}, \]
while for \(k=2p+2\),
\[ a=\frac{p+1}{(2p+1)^{2}}~\text{ and }b=\frac{(p+1)^{3}}{(2p+1)^{4}}. \]

MSC:

39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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