Liouville–Green–Olver Approximations for Complex Difference Equations

AbstractLiouville–Green (LG) orWKBapproximations for second-order lineardifferenceequations withcomplexcoefficients are obtained. Preciseboundsfor the error term in the asymptotic representation of theLGrecessive solution are given, and thedouble asymptoticnature, with respect to both,nand additional parameters, is shown; all this is in the spirit of F. W. J. Olver's rigorous work on theLGasymptotics for differential equations. Theholomorphiccharacter of such error terms, and hence of theLGbasis, is also established, when the coefficients of the difference equation are holomorphic. Qualitative properties, such as oscillation and growth of theLGbasis solutions, are displayed.Second-orderasymptotics with bounds is also obtained, and an application to three-term recurrences satisfied by certain orthogonal polynomials (a subclass of the Blumenthal-Nevai class), is made for illustration. The special case ofultrasphericalfunctions of the second kind is worked out in detail