Liouville-Green approximations for a class of linear oscillatory difference equations of the second order

AbstractAn asymptotic approximation theorem is proved for the solutions of linear oscillatory three-term recurrence equations in a certain class. This result represents a discrete analogue of the well-known Liouville-Green (or WKBJ) theorem rigorously proved by Olver for second-order linear differential equations. As in Olver's theorem, precise estimates are obtained for the error terms, and the double asymptotic nature of the approximation is made clear. Applications to certain orthogonal polynomials are also given