Representation of solutions of eight systems of difference equations via generalized Padovan sequences

We indicate that the systems of difference equations x_{n+۱}=f^{-۱}\big( af\left( p_{n-۱}\right)+bf\left( q_{n-۲}\right) \big) , \ \ y_{n+۱}=f^{-۱}\big( af\left( r_{n-۱}\right)+bf\left( s_{n-۲}\right) \big) ,\ \ n\in \mathbb{N}_{۰}, where the sequences p_{n}, q_{n}, r_{n}, s_{n} are some of the sequences x_{n} and y_{n}, f : D_f \longrightarrow \mathbb{R} be a ``۱-۱" continuous function on its domain D_f \subseteq \mathbb{R}, initial values x_{-j}, y_{-j}, j\in\{۰,۱,۲\} are arbitrary real numbers in D_f and the parameters a,b are arbitrary complex numbers, with b\neq ۰, can be solved in the closed form in terms of generalized Padovan sequences.

Keywords:

system of difference equations , solution of closed form , Padovan number