Nontrivial solutions of a higher-order rational difference equation. (English. Russian original) Zbl 1219.39007
Math. Notes 84, No. 5, 718-724 (2008); translation from Mat. Zametki 84, No. 5, 772-780 (2008).
Summary: We prove that, for any \(k\in \mathbb N\), the generalized Putnam difference equation
\[
x_{n+1}=\frac{x_n+x_{n-1}+\dots + x_{n-(k-1)}+x_{n-k}x_{n-(k+1)}}{x_n x_{n-1}+x_{n-2}+\dots +x_{n-(k+1)}},\quad n \in {\mathbb N}_0,
\]
with a positive solution has the asymptotics
\[
x_n=1+(k+1)e^{-\lambda^n}+(k+1)e^{- c \lambda^n}+o(e^{- c \lambda^n})
\]
for some \(c>1\) that depends on \(k\), where \(\lambda\) is a root of the polynomial \(P(\lambda)=\lambda^{k+2}-\lambda-1\) in the interval \((1,2)\). We use this result to prove that this equation has a positive solution that is not equal to one at the limit. Also, for the case \(k = 1\), we find all positive eventually equal to unity solutions to the equation.
MSC:
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
| 39A10 | Additive difference equations |
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