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

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Original languageEnglish
Pages (from-to)147-194
Journal / PublicationNumerische Mathematik
Issue number1
Publication statusPublished - Mar 2003


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 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.