Asymptotic expansion of the modified Lommel polynomials $h_{n,\nu }(x)$ and their zeros
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- by K. F. Lee and R. Wong PDF
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Abstract:
The modified Lommel polynomials satisfy the second-order linear difference equation \begin{equation*} h_{n+1,\nu }(x)-2(n+\nu ) x h_{n,\nu }(x)+ h_{n-1,\nu }(x)=0, \qquad n\geq 0, \end{equation*} 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.References
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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
- MR Author ID: 192744
- Email: rscwong@cityu.edu.hk
- Received by editor(s): July 6, 2012
- Received by editor(s) in revised form: December 31, 2012
- Published electronically: July 29, 2014
- Communicated by: Walter Van Assche
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3251735
Dedicated: Dedicated to the Lord