Uniform asymptotic expansions of the Pollaczek polynomials

Authors:
Yu-Qiu Zhao;Jian-Rong Zhou
Affiliations:
Department of Mathematics, ZhongShan University, GuangZhou, PR China;Department of Mathematics, ZhongShan University, GuangZhou, PR China
Venue:
Journal of Computational and Applied Mathematics - Special issue: International conference on mathematics and its application
Year:
2006

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Abstract

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials Pn(cos θ a, b). One is for θ ∈ (0, δ/√n], 0 a+b, in terms of elementary functions and in descending powers of √n. The other is for θ ∈ [δ/√n, π/2], in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of Pn (cos θ) in θ ∈ (0, π/2].