## 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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