@article{6afb00a33a9647aab75376a9d8217a4c, title = "Asymptotic expansions for second-order linear difference equations with a turning point", abstract = "A turning-point theory is developed for the second-order difference equation Pn+1 (x) - (Anx + Bn) Pn(x) + Pn-1(x) = 0, n = 1, 2, 3, ⋯, where the coefficients A n and Bn have asymptotic expansions of the form A n ∼ n-θ Σs=0 ∞ αs/ns and Bn ∼ Σ s=0 ∞ βs/ns, 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.", author = "Z. Wang and R. Wong", year = "2003", month = mar, doi = "10.1007/s00211-002-0416-y", language = "English", volume = "94", pages = "147--194", journal = "Numerische Mathematik", issn = "0029-599X", publisher = "Springer", number = "1", }