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Asymptotic behavior of the Pollaczek polynomials and their zeros. (English) Zbl 0859.41025
The Pollaczek polynomials are represented by means of the Cauchy integral that follows from the generating function. The saddle point method is used to derive an expansion in terms of Airy functions. The paper gives a detailed discussion of the complicated saddle point analysis, the conformal mapping of the phase function to a cubic polynomial, the leading terms of the expansion, and the local (non-uniform) behavior that can be derived from the Airy-type expansion. In the discussion on the zeros of the polynomials the authors compare the approximation for the zeros with an earlier result obtained by M. E. H. Ismail [SIAM J. Math. Anal. 25, No. 2, 462-473 (1994; Zbl 0805.33005)], and they give corrections to this result.
MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C45Orthogonal polynomials and functions of hypergeometric type