Numerische Mathematik

, Volume 94, Issue 1, pp 147–194 | Cite as

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

  • Z. Wang
  • R. Wong

Summary.

 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), xℝ.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Z. Wang
    • 1
  • R. Wong
    • 1
  1. 1.Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; e-mail: mawong@cityu.edu.ukGB

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