Asymptotic expansions for second-order linear difference equations with a turning point

A turning-point theory is developed for the second-order difference equation P_n+1 (x) - (A_nx + B_n) P_n(x) + P_n-1(x) = 0, n = 1, 2, 3, ⋯, where the coefficients A _n and B_n have asymptotic expansions of the form A _n ∼ n^-θ Σ_s=0 ^∞ α_s/n^s and B_n ∼ Σ _s=0 ^∞ β_s/n^s, θ ≠ 0 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 exp(-x⁴), x ∈ ℝ.