ABSTRACT
This chapter is the heart of this book. In this chapter we present the known results on each of the 225 special cases of the third-order rational difference equation
xn+1 = α+ βxn + γxn−1 + δxn−2 A+Bxn + Cxn−1 +Dxn−2
, n = 0, 1, . . . (5.0.1)
with nonnegative parameters α, β, γ, δ, A,B,C,D and with arbitrary nonnegative initial conditions x−2, x−1, x0, such that the denominator is always positive. In several special cases we also present some new results and pose some
open problems and conjectures on the character of their solutions. Whenever we can, we extend the results to the general (k + 1)st-order
rational difference equation
xn+1 = α+
A+ ∑k
i=0Bixn−i , n = 0, 1, . . . . (5.0.2)
The ultimate goal for the reader is to generalize each case in this chapter to the most general functional equation
xn+1 = f(xn, . . . , xn−k), n = 0, 1, . . . .
