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

    https://doi.org/10.1142/9789814656054_0044Cited by:0 (Source: Crossref)
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    Abstract:

    A turning-point theory is developed for the second-order difference equation

    Pn+1 − (Anx + Bn)Pn(x) + Pn−1 = 0, n = 1, 2, 3, ··· ,
    where the coefficients An and Bn have asymptotic expansions of the form
    , 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