No Access

Asymptotics of the Charlier polynomials via difference equation methods

Xiao-Min Huang

School of Applied Mathematics, Guangdong University of Technology, Guangzhou, P. R. China

Department of Mathematics, City University of Hong Kong, Hong Kong

Search for more papers by this author

Yu Lin

Department of Mathematics, South China University of Technology, Guangzhou, P. R. China

Search for more papers by this author

, and

Yu-Qiu Zhao

Department of Mathematics, Sun Yat-sen University, Guangzhou, P. R. China

E-mail Address: stszyq@mail.sysu.edu.cn

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S021953052050013XCited by:2 (Source: Crossref)

Abstract

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained, respectively, in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.

Keywords:

AMSC: 41A60, 39A10, 33C45

References

1. R. Askey, Review of the book An Introduction to Orthogonal Polynomials, by T. S. Chihara, Gordon & Breach, New York/London/Paris, 1978, J. Approx. Theory 43 (1985) 394–395. Google Scholar
2. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin and P. D. Miller, Discrete Orthogonal Polynomials, Asymptotics and Applications, Annals Mathematics Studies, Vol. 164 (Princeton University Press. Princeton and Oxford, 2007). Google Scholar
3. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Reprint of the 1978 original (Springer-Verlag, New York, 1999). Google Scholar
4. R. Bo and R. Wong, Uniform asymptotic expansion of Charlier polynomials, Methods Appl. Anal. 1 (1994) 294–313. Google Scholar
5. L.-H. Cao and Y.-T. Li, Linear difference equations with a transition point at the origin, Anal. Appl. 12 (2014) 75–106. Link, Web of Science, Google Scholar
6. T. S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978). Google Scholar
7. O. Costin and R. Costin, Rigorous WKB for finite-order linear recurrence relations with smooth coefficients, SIAM J. Math. Anal. 27 (1996) 110–134. Web of Science, Google Scholar
8. D. Dai, M. E. H. Ismail and X.-S. Wang, Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems, Constr. Approx. 40 (2014) 61–104. Web of Science, Google Scholar
9. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, Vol. 3 (New York University, 1999). Google Scholar
10. T. M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) 93–133. Web of Science, Google Scholar
11. J. S. Geronimo, WKB and turning point theory for second order difference equations: External fields and strong asymptotics for orthogonal polynomials, preprint (2009), arXiv:0905.1684. Google Scholar
12. W. M. Y. Goh, Plancherel-Rotach asymptotics for the Charlier polynomials, Constr. Approx. 14 (1998) 151–168. Web of Science, Google Scholar
13. X.-M. Huang, The difference equation method to asymptotic expansion of orthogonal polynomials and confluent Heun equation, Ph.D. Thesis, Sun Yat-sen University (2018). Google Scholar
14. X.-M. Huang, L.-H. Cao and X.-S. Wang, Asymptotic expansion of orthogonal polynomials via difference equations, J. Approx. Theory 239 (2019) 29–50. Web of Science, Google Scholar
15. A. B. J. Kuijlaars and W. Van Assche, The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients, J. Approx. Theory 99 (1999) 167–197. Web of Science, Google Scholar
16. M. Maejima and W. Van Assche, Probabilistic proofs of asymptotic formulas for some classical polynomials, Math. Proc. Cambridge Philos. Soc. 97 (1985) 499–510. Web of Science, Google Scholar
17. F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010). Google Scholar
18. C.-H. Ou and R. Wong, The Riemann-Hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes, Anal. Appl. 8 (2010) 247–286. Link, Web of Science, Google Scholar
19. G. Szegő, Orthogonal Polynomials, 4th edn (American Mathematical Society, Providence, RI, 1975). Google Scholar
20. W. Van Assche and J. S. Geronimo, Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients, Rocky Mountain J. Math. 19 (1989) 39–49. Web of Science, Google Scholar
21. X.-S. Wang and R. Wong, Asymptotics of orthogonal polynomials via recurrence relations, Anal. Appl. 10 (2012) 215–235. Link, Web of Science, Google Scholar
22. Z. Wang and R. Wong, Uniform asymptotic expansion of $J_{ν} (ν a)$ via a difference equation, Numer. Math. 91 (2002) 147–193. Web of Science, Google Scholar
23. Z. Wang and R. Wong, Asymptotic expansions for second-order linear difference equations with a turning point, Numer. Math. 94 (2003) 147–194. Web of Science, Google Scholar
24. Z. Wang and R. Wong, Linear difference equations with transition points, Math. Comp. 74 (2005) 629–653. Web of Science, Google Scholar
25. R. Wong, Asymptotics of linear recurrences, Anal. Appl. 12 (2014) 463–484. Link, Web of Science, Google Scholar
26. R. Wong and H. Li, Asymptotic expansions for second-order linear difference equations, J. Comput. Appl. Math. 41 (1992) 65–94. Web of Science, Google Scholar