Authors: Stević, Stevo 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Nontrivial solutions of a higher-order rational difference equation
Journal: Mathematical Notes
Volume: 84
Issue: 5-6
First page: 718
Last page: 724
Issue Date: 1-Dec-2008
Rank: M23
ISSN: 0001-4346
DOI: 10.1134/S0001434608110138
We prove that, for every k ∈ ℕ, the following generalization of the Putnam difference equation xn+1 = xn + xn-1 +...+ xn-(k-1) + xn-kxn-(k+1)/x nxn-1 + xn-2 + ... + xn-(k+1) n ∈ ℕ0 has a positive solution with the following asymptotics xn = 1 + (k+1)e-cλn + o(e e -cλn) for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ k+2 - λ - 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: Asymptotic | Difference equation | Nonlinear solution | Putnam difference equation
Publisher: Springer Link

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