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On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. (English) Zbl 0971.41016
An infinite asymptotic expansion is derived for the Meixner-Pollaczek polynomials M n (nα;δ,n) as n, which holds uniformly for -MαM, where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α n,s denotes the sth zero of M n (nα;δ,η), counted from the right, and if α ˜ n,s denotes sth zero counted form the left, then for each s, three-term asymptotic approximations are obtained for both α n,s and α ˜ n,s as n.
Reviewer: F.Pérez Acosta
MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C45Orthogonal polynomials and functions of hypergeometric type