Asymptotic expansions for second-order linear difference equations with a turning point
A turning-point theory is developed for the second-order difference equation
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), xℝ.
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© Springer-Verlag Berlin Heidelberg 2003