Iranian Journal of Mathematical Sciences and Informatics مجله علوم ریاضی و انفورماتیک ایرانیان IJMSI Basic Sciences http://ijmsi.ir 1 admin 1735-4463 2008-9473 8 7 14 8888 13 en jalali 1397 2 1 gregorian 2018 5 1 13 1 online 1 fulltext
en 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/(&plusmn;1 + x(n)), n &isin; N_0. The form of solution of this equation, however, was first obtained in  but through induction principle. Then, with the solution of the above equation at hand, we prove a case of Sroysang&rsquo;s conjecture (2013)  i.e., given a fixed positive integer k, we verify the validity of the following claim: lim x&rarr;&infin; f(x + k)/f(x) = &phi;, where &phi; = (1 + &radic;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 &isin; 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