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

DOI:
https://doi.org/10.1090/S0002-9939-10-10301-3

Published electronically:
March 4, 2010

MathSciNet review:
2607881

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Abstract | References | Similar Articles | Additional Information

Abstract: The Tricomi-Carlitz polynomials satisfy the second-order linear difference equation

**91**(2002) and

**94**(2003)]. The result holds uniformly in regions containing the critical values , where .

**[1]**L. Carlitz, On some polynomials of Tricomi,*Boll. Un. Mat. Ital.*,**13**, 58-64, 1958. MR**0103303 (21:2078)****[2]**W.M.Y. Goh and J. Wimp, On the asymptotics of the Tricomi-Carlitz polynomials and their zero distribution. I,*SIAM J. Math. Anal.*,**25**, 420-428, 1994. MR**1266567 (95b:42025)****[3]**W.M.Y. Goh and J. Wimp, The zero distribution of the Tricomi-Carlitz polynomials. Approximation theory and applications,*Comput. Math. Appl.*,**33**, 119-127, 1997. MR**1442066 (99e:33006)****[4]**J.L. López and N.M. Temme, Approximation of orthogonal polynomials in terms of Hermite polynomials. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II,*Methods Appl. Anal.*,**6**, 131-146, 1999. MR**1803886 (2001m:33013)****[5]**F.W.J. Olver,*Asymptotics and Special Functions*, Academic Press, New York, 1974. (Reprinted by A. K. Peters, Ltd., Wellesley, 1997.) MR**0435697 (55:8655)****[6]**G. Szegö,*Orthogonal Polynomials*, 4th edition,*Amer. Math. Soc. Colloq. Publ.*, 23, Amer. Math. Soc., Providence, RI, 1975. MR**0372517 (51:8724)****[7]**F.G. Tricomi, A class of non-orthogonal polynomials related to those of Laguerre,*J. Analyse Math.*,**13**, 209-231, 1951. MR**0051351 (14:466e)****[8]**Z. Wang and R. Wong, Uniform asymptotic expansion of via a difference equation,*Numer. Math.*,**91**, 147-193, 2002. MR**1896091 (2003g:33008)****[9]**Z. Wang and R. Wong, Asymptotic expansions for second-order linear difference equations with a turning point,*Numer. Math.*,**94**, 147-194, 2003. MR**1971216 (2004c:39012)****[10]**R. Wong,*Asymptotic Approximations of Integrals*, Academic Press, Boston, 1989. MR**1016818 (90j:41061)**

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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:
https://doi.org/10.1090/S0002-9939-10-10301-3

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.