Iranian Journal of Mathematical Sciences and Informatics
مجله علوم ریاضی و انفورماتیک ایرانیان
IJMSI
Basic Sciences
http://ijmsi.ir
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admin
1735-4463
2008-9473
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jalali
1397
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gregorian
2018
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On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang’s Conjecture
تخصصي
Special
پژوهشي
Research paper
<p>The purpose of this paper is twofold. First we derive theoretically, using appropriate transformation on x(n), the closed-form solution of the nonlinear difference equation x(n+1) = 1/(±1 + x(n)), n ∈ N_0. The form of solution of this equation, however, was first obtained in [10] but through induction principle. Then, with the solution of the above equation at hand, we prove a case of Sroysang’s conjecture (2013) [9] i.e., given a fixed positive integer k, we verify the validity of the following claim: lim x→∞ f(x + k)/f(x) = φ, where φ = (1 + √5)/2 denotes the well-known golden ratio and the real valued function f on R satisfies the functional equation f(x + 2k) =f(x + k) + f(x) for every x ∈ R. We complete the proof of the conjecture by giving out an entirely different approach for the other case.</p>
Golden ratio, Fibonacci functional equation, Horadam functional equation, convergence.
139
151
http://ijmsi.ir/browse.php?a_code=A-10-1534-1&slc_lang=en&sid=1
J. F.
Rabago
jfrabago@gmail.com
`10031947532846005296`

10031947532846005296
Yes
University of the Philippines Baguio