Elaydi Abstract Poincar -Perron's Theorem marks the beginning of the asymptotic theory of scalar nonautonomous difference equations. However, this theorem is of limited use due to two reasons. The first is the requirement that all characteristic roots of the limiting equation have distinct moduli. The second reason is that the theorem does not provide explicit asymptotic representations of solutions of difference equations. Levinson's Theorem or rather its discrete analogue addressed the second deficiency by providing the desirable asymptotic representation of solutions of difference equations. However, Levinson's Theorem puts a severe restriction on the coefficients of the difference equation in question since it requires that they are summable. The latter condition makes the applicability of the the theorem rather limited, particulaply in orthogonal polynomials, combinatorics, special functions, and continued fractions. The proposed research is directed toward addressing the deficiencies in the above two important theorems. It will attempt to complete and unify asymptotic theory using a unified approach via the theory of dichotomy. Difference equations are one of the basic mathematical tools of computation. There is hardly a computational task which does not rely on recursive techniques, at one time or another. The widespread use of difference equations can be ascribed to their intrinsic constructive quality, and the great ease with which they are amenable to mechanization. On the other hand, like most recursive processes, difference equations are susceptible to error growth. If conditions are unfavorable, the resulting propagation of error may or will be disastrous. Hence the need to investigate the asymptotic and qualitative behavior of solutions of difference equations. This is of paramount importance since difference equations occur prominently in many branches of applied mathematics, Economics, population dynamics, Chaos Theory, mathematical physics, etc. Asymptotics is even more important in Computer Science where it is used to rate algorithms and how fast and accurate they are.
