Asymptotic expansions for second-order linear difference equations with a turning point
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A turning-point theory is developed for the second-order difference equation
where the coefficients A n and B n 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(−x 4 ), xℝ.
KeywordsReal Number Asymptotic Expansion Difference Equation Turning Point Orthogonal Polynomial
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© Springer-Verlag Berlin Heidelberg 2003