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Asymptotic expansion of the modified Lommel polynomials $ h_{n,\nu}(x)$ and their zeros


Authors: K. F. Lee and R. Wong
Journal: Proc. Amer. Math. Soc. 142 (2014), 3953-3964
MSC (2010): Primary 41A60, 39A10; Secondary 33C45
DOI: https://doi.org/10.1090/S0002-9939-2014-12134-4
Published electronically: July 29, 2014
MathSciNet review: 3251735
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Abstract: The modified Lommel polynomials satisfy the second-order linear difference equation

$\displaystyle h_{n+1,\nu }(x)-2(n+\nu )\,x\,h_{n,\nu }(x)+ h_{n-1,\nu }(x)=0, \qquad n\geq 0,$    

with initial values $ h_{-1,\nu }(x)=0$ and $ h_{0,\nu }(x)=1$, where $ x$ is a real variable and $ \nu $ is a fixed positive parameter. An asymptotic expansion, as $ n \to \infty $, is derived for these polynomials by using a 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 the infinite interval $ 0\leq x<\infty $, containing the critical value $ x=1/{N}$, where $ N=n+\nu $. Behavior of the zeros of these polynomials is also studied.

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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/S0002-9939-2014-12134-4
Keywords: Modified Lommel polynomials, second-order linear difference equations, uniform asymptotic expansions, Airy function
Received by editor(s): July 6, 2012
Received by editor(s) in revised form: December 31, 2012
Published electronically: July 29, 2014
Dedicated: Dedicated to the Lord
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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