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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniform asymptotic expansions of the Tricomi-Carlitz polynomials


Authors: K. F. Lee and R. Wong
Journal: Proc. Amer. Math. Soc. 138 (2010), 2513-2519
MSC (2010): Primary 41A60, 39A10; Secondary 33C45
Published electronically: March 4, 2010
MathSciNet review: 2607881
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Abstract: The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation

$\displaystyle (n+1)f_{n+1}^{(\alpha)}(x)-(n+\alpha) x f_n^{(\alpha)}(x)+ f_{n-1}^{(\alpha)}(x)=0, \qquad\qquad n\geq1, $

with initial values $ f_0^{(\alpha)}(x)=1$ and $ f_1^{(\alpha)}(x)=\alpha x$, where $ x$ is a real variable and $ \alpha$ is a positive parameter. An asymptotic expansion is derived for these polynomials by using the turning-point theory for three-term recurrence relations developed by Wang and Wong [Numer. Math. 91(2002) and 94(2003)]. The result holds uniformly in regions containing the critical values $ x=\pm2/\sqrt{\nu}$, where $ \nu=n+2\alpha-1/2$.


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Additional Information

K. F. Lee
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Email: charleslkf8571@gmail.com

R. Wong
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Email: rscwong@cityu.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10301-3
PII: S 0002-9939(10)10301-3
Keywords: Tricomi-Carlitz polynomials, uniform asymptotic expansion, difference equation
Received by editor(s): June 23, 2009
Received by editor(s) in revised form: November 6, 2009
Published electronically: March 4, 2010
Dedicated: Dedicated to the Lord
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.