Liouville-Green-Olver approximations for complex difference equations

Liouville-Green (LG) or WKB approximations for second-order linear differen ce equations with complex coefficients are obtained. Precise bounds fbr the error term in the asymptotic representation of the LG recessive solution a re given, and the double asymptotic nature, with respect to both, n and add itional 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 holomo rphic character of such error terms, and hence of the LG basis, is also est ablished, when the coefficients of the difference equation are holomorphic. Qualitative properties, such as oscillation and growth of the LG basis sol utions, are displayed. Second-order asymptotics with bounds is also obtaine d, and an application to three-term recurrences satisfied by certain orthog onal polynomials (a subclass of the Blumenthal-Nevai class), is made for il lustration. The special case of ultraspherical functions of the second kind is worked out in detail. (C) 1999 Academic Press.