Some representations of the general solution to a difference equation of additive type

Abstract

Abstract The general solution to the difference equation $$x_{n+1}=\frac {ax_{n}x_{n-1}x_{n-2}+bx_{n-1}x_{n-2}+cx_{n-2}+d}{x_{n}x_{n-1}x_{n-2}},\quad n\in\mathbb{N}_{0}, $$ x n + 1 = a x n x n − 1 x n − 2 + b x n − 1 x n − 2 + c x n − 2 + d x n x n − 1 x n − 2 , n ∈ N 0 , where $a, b, c\in\mathbb{C}$ a , b , c ∈ C , $d\in\mathbb{C}\setminus\{0\}$ d ∈ C ∖ { 0 } , is presented by using the coefficients, the initial values $x_{-j}$ x − j , $j=\overline{0,2}$ j = 0 , 2 ‾ , and the solution to the difference equation $$y_{n+1}=ay_{n}+by_{n-1}+cy_{n-2}+dy_{n-3}, \quad n\in\mathbb{N}_{0}, $$ y n + 1 = a y n + b y n − 1 + c y n − 2 + d y n − 3 , n ∈ N 0 , satisfying the initial conditions $y_{-3}=y_{-2}=y_{-1}=0$ y − 3 = y − 2 = y − 1 = 0 , $y_{0}=1$ y 0 = 1 . The representation complements known ones of the general solutions to the corresponding difference equations of the first and second order. Besides, the general representation formula is investigated in detail and refined by using the roots of the characteristic polynomial $$P_{4}(\lambda )=\lambda ^{4}-a\lambda ^{3}-b\lambda ^{2}-c\lambda -d $$ P 4 ( λ ) = λ 4 − a λ 3 − b λ 2 − c λ − d of the linear equation. The following cases are considered separately: (1) all the roots of the polynomial are distinct; (2) there is a unique double root of the polynomial; (3) there is a triple root of the polynomial and one simple; (4) there is a quadruple root of the polynomial; (5) there are two distinct double roots of the polynomial.