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 Mat. Zametki, 2008, Volume 84, Issue 5, Pages 772–780 (Mi mz6360)

Nontrivial Solutions of a Higher-Order Rational Difference Equation

S. Stević

Mathematical Institute, Serbian Academy of Sciences and Arts

Abstract: We prove that, for every $k\in\mathbb N$, the following generalization of the Putnam difference equation
$$x_{n+1}=\frac{x_n+x_{n-1}+…+x_{n-(k-1)}+x_{n-k}x_{n-(k+1)}} {x_nx_{n-1}+x_{n-2}+…+x_{n-(k+1)}} ,\qquad n\in\mathbb N_0,$$
has a positive solution with the following asymptotics
$$x_n=1+(k+1)e^{-\lambda^n}+(k+1)e^{-c\lambda^n}+o(e^{-c\lambda^n})$$
for some $c>1$ depending on $k$, and where $\lambda$ is the root of the polynomial $P(\lambda)=\lambda^{k+2}-\lambda-1$ belonging to the interval $(1,2)$. Using this result, we prove that the equation has a positive solution which is not eventually equal to $1$. Also, for the case $k=1$, we find all positive eventually equal to unity solutions to the equation.

Keywords: difference equation, nonlinear solution, asymptotic, Putnam difference equation

DOI: https://doi.org/10.4213/mzm6360

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English version:
Mathematical Notes, 2008, 84:5, 718–724

Bibliographic databases:

UDC: 512.628.4

Citation: S. Stević, “Nontrivial Solutions of a Higher-Order Rational Difference Equation”, Mat. Zametki, 84:5 (2008), 772–780; Math. Notes, 84:5 (2008), 718–724

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz6360
• https://doi.org/10.4213/mzm6360
• http://mi.mathnet.ru/eng/mz/v84/i5/p772

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