Asymptotics of linear recurrences

Abstract

In this paper, we review the asymptotic results that are now available for second-order linear difference equations containing a parameter (or, equivalently, three-term recurrence relations). These include asymptotic expansions for solutions to these equations when the parameter is fixed or varying in an interval containing a turning point or a transition point. Also presented is a method for deriving asymptotic approximations for solutions when the initial values are given. These results are particularly useful when a given system of orthogonal polynomials (i) does not satisfy any second-order differential equation, (ii) does not have any integral representation, and (iii) is not even associated with a unique (or any) weight function.