Authors: Stević, Stevo 
Title: Some representations of the general solution to a difference equation of additive type
Journal: Advances in Difference Equations
Volume: 2019
Issue: 1
Issue Date: 1-Dec-2019
Rank: M21а
ISSN: 1687-1847
DOI: 10.1186/s13662-019-2365-0
Abstract: 
The general solution to the difference equation xn+1=axnxn−1xn−2+bxn−1xn−2+cxn−2+dxnxn−1xn−2,n∈N0, where a, b, c∈ C, d∈ C∖ { 0 } , is presented by using the coefficients, the initial values x−j, j= 0 , 2 ‾ , and the solution to the difference equation yn+1=ayn+byn−1+cyn−2+dyn−3,n∈N0, satisfying the initial conditions y− 3= y− 2= y− 1= 0 , y= 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 P4(λ) = λ4− aλ3− bλ2− cλ− 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.
Keywords: Linear difference equation | Representation of solutions | Solvable difference equation | Third-order difference equation
Publisher: Springer Link
Project: Modulation of intracellular energy balance-controlling signalling pathways in therapy of cancer and neuro-immuno-endocrine disorders 
Development of new information and communication technologies, based on advanced mathematical methods, with applications in medicine, telecommunications, power systems, protection of national heritage and education 

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