Classification of neutral difference equations of any order with respect to the asymptotic behavior of their solutions

The asymptotic behavior of the solutions to a neutral difference equation of the formΔmx(n)+q(n)x(τ(n))+p(n)x(σ(n))=0,m∈N,n⩾0,where τ(n) is a general retarded argument, σ(n) is a general deviated argument, p(n)n⩾0 and q(n)n⩾0 are sequences of real numbers, Δ denotes the forward difference operator Δx(n)=x(n+1)-x(n), and Δj denotes the jth forward difference operator Δjx(n)=ΔΔj-1x(n) for j=2,3,…,m, is studied. Examples illustrating the results are also given.