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Zhou JR, Zhao YQ. Uniform asymptotics of the Pollaczek polynomials via the Riemann–Hilbert approach. Proc Math Phys Eng Sci 2008. [DOI: 10.1098/rspa.2007.0385] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The Pollaczek weight is an example of the non-Szegö class. In this paper, we investigate the asymptotics of the Pollaczek polynomials via the Riemann–Hilbert approach. In the analysis, the original endpoints ±1 of the orthogonal interval are shifted to the Mhaskar–Rakhmanov–Saff numbers
α
n
and
β
n
. It is also shown, by analysing the singularities of the
ϕ
-function, that the endpoint parametrices constructed in terms of the Airy function are bound to be local. Asymptotic approximations are obtained in overlapping regions that cover the whole complex plane. The approximations, some special values and the leading and recurrence coefficients are compared with the known results.
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Affiliation(s)
- Jian-Rong Zhou
- Department of Mathematics, Zhongshan UniversityGuangzhou 510275, People's Republic of China
| | - Yu-Qiu Zhao
- Department of Mathematics, Zhongshan UniversityGuangzhou 510275, People's Republic of China
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