Summary.
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.
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Received February 21, 2002 / Revised version received April 8, 2002 / Published online October 29, 2002
Mathematics Subject Classification (1991): 41A60, 39A10, 33C45
The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1132 | 00P)
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Wang, Z., Wong, R. Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94, 147–194 (2003). https://doi.org/10.1007/s00211-002-0416-y
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DOI: https://doi.org/10.1007/s00211-002-0416-y