Liouville-Green-Olver approximations for complex difference equations

Liouville-Green (LG) or WKB approximations for second-order linear difference equations with complex coefficients are obtained. Precise bounds for the error term in the asymptotic representation of the LG recessive solution are given, and the double asymptotic nature, with respect to both, n and additional parameters, is shown; all this is in the spirit of F. W. J. Olver's rigorous work on the LG asymptotics for differential equations. The holomorphic character of such error terms, and hence of the LG basis, is also established, when the coefficients of the difference equation are holomorphic. Qualitative properties, such as oscillation and growth of the LG basis solutions, are displayed. Second-order asymptotics 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 of ultraspherical functions of the second kind is worked out in detail. © 1999 Academic Press.