Authors’ summary: A turning-point theory is developed for the second-order difference equation
where the coefficients and have asymptotic expansions of the form
being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight , .