Abstract
General solutions to four classes of nonlinear difference equations and some of their representations
Highlights
In this paper N denotes the set of all positive integers, N0 = N ∪ {0}, Z the set of all integers, R the set of all real numbers, whereas the set of all complex numbers is denoted by C.One of the oldest problems regarding difference equations is their solvability
There was said that difference equations (1.8), (1.11), (1.14) and (1.17) can be treated as in the previous cases, where some closed-form formulas for the general solutions to two difference equations are proved by induction
By using the method of transformation, which is based on some suitably chosen changes of variables which transform original difference equations to some known solvable ones, we show that difference equations (1.20)–(1.23) are solvable in closed form
Summary
There are some recent papers which give representations of solutions to very special cases of some solvable difference equations in terms of the Fibonacci sequence. There was said that difference equations (1.8), (1.11), (1.14) and (1.17) can be treated as in the previous cases, where some closed-form formulas for the general solutions to two difference equations are proved by induction This means that [7] does not give any explanation for getting any of the closed-form formulas mentioned there including the formulas (1.9), (1.10), (1.12), (1.13), (1.15), (1.16), (1.18) and (1.19), that is, [7] does not present any constructive method for getting them. The representation formulas (1.9), (1.10), (1.15) and (1.16), are considerably extended, by presenting infinitely many other related representations to the general solution to equations (1.20) and (1.22), respectively This shows that the representation in terms of the Fibonacci sequence is one of many possible representations, and that more or less the sequence is chosen arbitrary. From the point of view of solvability, the choice of the Fibonacci sequence in such representations does not have some advantages with the respect to the other ones
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