Abstract
Meixner polynomials mn(x; β, c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are formula Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for mn(nα β, c) as n → ∞ One holds uniformly for 0 > ɛ ≤ α > 1 + a, and the other holds uniformly for 1 - b > α > M > ∞, where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases. © 2016 by World Scientific Publishing Co. Ptc. Ltd.
| Original language | English |
|---|---|
| Title of host publication | Selected Works Of Roderick S. C. Wong, The (In 3 Volumes) |
| Publisher | World Scientific Publishing Co. Pte Ltd |
| Pages | 568-605 |
| ISBN (Print) | 9789814656054 |
| DOIs | |
| Publication status | Published - 5 Aug 2015 |
Bibliographical note
Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].Funding
The authors would like to thank the referee for a careful reading of this paper and several helpful suggestions. This research was partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A7359.
Research Keywords
- Meixner polynomials
- Parabolic cylinder function
- Steepest descent method
- Uniform asymptotic expansions
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