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An Infinite Asymptotic Expansion for the Extreme Zeros of the Pollaczek Polynomials
Author(s) -
Zhou JianRong,
Zhao YuQiu
Publication year - 2007
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2007.00373.x
Subject(s) - mathematics , asymptotic expansion , airy function , mathematical analysis , transformation (genetics) , taylor series , constant (computer programming) , function (biology) , analytic function , expression (computer science) , interval (graph theory) , combinatorics , biochemistry , chemistry , evolutionary biology , biology , computer science , gene , programming language
In this paper, we first establish an integral expression for the Pollaczek polynomials P n ( x ; a , b ) from a generating function. By applying a canonical transformation to the integral and carrying out a detailed analysis of the integrand, we derive a uniform asymptotic expansion for P n (cosθ; a , b ) in terms of the Airy function and its derivative, in descending powers of n . The uniformity is in an interval next to the turning point , with M being a constant. The coefficients of the expansion are analytic functions of a parameter that depends only on t where , and not on the large parameter n . From the expansion of the polynomials we obtain an asymptotic expansion in powers of n −1/3 for the largest zeros. As a special case, a four‐term approximation is provided for comparison and illustration. The method used in this paper seems to be applicable to more general situations.