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Zbl 0958.33004
López, José L.; Temme, Nico M.
Approximations of orthogonal polynomials in terms of Hermite polynomials. (English)
[J] Methods Appl. Anal. 6, No.2, 131-146 (1999). ISSN 1073-2772

It is known that the asymptotics of Laguerre polynomials $L_n^\alpha(x)$ (or Gegenbauer polynomials $C_n^\gamma(x)$) can be expressed in terms of Hermite polynomials for large values of the order parameter $\alpha$ (or $\gamma$). This paper gives a uniform approach, based on generating functions, to derive such asymptotic expressions, not only for these but also for other classes of orhogonal polynomials. The details are worked out for Gegenbauer, Laguerre, Jacobi and Tricomi-Carlitz polynomials. From these asymptotics, estimates for the zeros of the polynomials can be obtained in terms of the zeros of Hermite polynomials. Also this aspect is worked out for the above mentioned classes.
[Adhemar Bultheel (Leuven)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
41A60 Asymptotic problems in approximation
30C15 Zeros of polynomials, etc. (one complex variable)
41A10 Approximation by polynomials

Keywords: asymptotics; orthogonal polynomials; zero distribution

Cited in: Zbl 1168.33309

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