zbMATH — the first resource for mathematics

Global behavior of solutions for a difference equation of third order. (English) Zbl 1369.39010
Summary: n this paper we consider the third-order rational difference equation \[ y_{n+1}\frac{y_{n-1}-y_{n-2}}{1-ay_{n-2}},n\in\mathbb N_0, \] where parameter \(a\) is a nonzero real number and the initial values \(y_{-2}\), \(y_{-1}\), \(y_0\in\mathbb R\backslash \{\frac{1}{a}\}\). We here determine both the forms and the global behavior a of the solutions of the above equation. Also, we show that the solutions are associated with Padovan numbers which contribute to explain the global behavior of the solutions.
39A23 Periodic solutions of difference equations
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
39A10 Additive difference equations
Full Text: Link