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Zbl 1030.39016
Wang, Z.; Wong, R.
Asymptotic expansions for second-order linear difference equations with a turning point. (English)
[J] Numer. Math. 94, No.1, 147-194 (2003). ISSN 0029-599X; ISSN 0945-3245/e

Authors' summary: A turning-point theory is developed for the second-order difference equation $$ P_{n+1}(x)-(A_nx+B_n)P_{n}(x) +P_{n-1}(x)=0,\quad n=1,2,3,\dots $$ where the coefficients $A_n$ and $B_n$ have asymptotic expansions of the form $$ A_n\sim n^{-\theta}\sum_{s=0}^{\infty} \frac{\alpha_s}{n^s}\quad \text { and }\quad B_n\sim \sum_{s=0}^{\infty} \frac{\beta_s}{n^s}, $$ $\theta\neq 0$ 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 $\exp(-x^4)$, $x\in\Bbb R$.
[Eduardo Liz (Vigo)]

MSC 2000:: *39A11 Stability of difference equations
33C10 Cylinder functions, etc.
41A60 Asymptotic problems in approximation

Keywords: turning-point theory; asymptotic expansions; second-order linear difference equations; orthogonal polynomials; Airy functions; three-term recurrence relation

Cited in: Zbl 1202.41032

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