Comptes Rendus
Mathematical Analysis
Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters
[Analyse asymptotique uniforme des polynômes de Meixner–Pollaczek avec des paramètres variables]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1031-1035.

Dans cette Note, nous effectuons une analyse asymptotique uniforme des polynômes de Meixner–Pollaczek Pn(λn)(z;ϕ) avec un paramètre λn=(n+12)A lorsque n, où A>0 est une constante. Des développements asymptotiques en termes de fonctions paraboliques cylindriques et de fonctions élémentaires sont obtenus de manière uniforme en z dans deux régions qui recouvrent tout le plan complexe.

In this Note, we study the uniform asymptotics of the Meixner–Pollaczek polynomials Pn(λn)(z;ϕ) with varying parameter λn=(n+12)A as n, where A>0 is a constant. Uniform asymptotic expansions in terms of parabolic cylinder functions and elementary functions are obtained for z in two overlapping regions which together cover the whole complex plane.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.08.020

Jun Wang 1 ; Weiyuan Qiu 1 ; Roderick Wong 2

1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
@article{CRMATH_2011__349_19-20_1031_0,
     author = {Jun Wang and Weiyuan Qiu and Roderick Wong},
     title = {Uniform asymptotics for {Meixner{\textendash}Pollaczek} polynomials with varying parameters},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1031--1035},
     publisher = {Elsevier},
     volume = {349},
     number = {19-20},
     year = {2011},
     doi = {10.1016/j.crma.2011.08.020},
     language = {en},
}
TY  - JOUR
AU  - Jun Wang
AU  - Weiyuan Qiu
AU  - Roderick Wong
TI  - Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1031
EP  - 1035
VL  - 349
IS  - 19-20
PB  - Elsevier
DO  - 10.1016/j.crma.2011.08.020
LA  - en
ID  - CRMATH_2011__349_19-20_1031_0
ER  - 
%0 Journal Article
%A Jun Wang
%A Weiyuan Qiu
%A Roderick Wong
%T Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters
%J Comptes Rendus. Mathématique
%D 2011
%P 1031-1035
%V 349
%N 19-20
%I Elsevier
%R 10.1016/j.crma.2011.08.020
%G en
%F CRMATH_2011__349_19-20_1031_0
Jun Wang; Weiyuan Qiu; Roderick Wong. Uniform asymptotics for Meixner–Pollaczek polynomials with varying parameters. Comptes Rendus. Mathématique, Volume 349 (2011) no. 19-20, pp. 1031-1035. doi : 10.1016/j.crma.2011.08.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.08.020/

[1] J. Baik; T.M. Suidan Random matrix central limit theorems for nonintersecting random walks, Ann. Probab., Volume 35 (2007), pp. 1807-1834

[2] Y. Chen; M.E.H. Ismail Asymptotics of the extreme zeros of the Meixner–Pollaczek polynomials, J. Comput. Appl. Math., Volume 82 (1997), pp. 59-78

[3] P. Deift; X. Zhou A steepest descent method for oscillatory Riemann–Hilbert problems, asymptotic for the mKdV equation, Ann. of Math., Volume 137 (1993) no. 2, pp. 295-368

[4] A.S. Fokas; A.R. Its; A.V. Kitaev The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys., Volume 147 (1992), pp. 395-430

[5] I.V. Krasovsky Asymptotic distribution of zeros of polynomials satisfying difference equations, J. Comput. Appl. Math., Volume 150 (2003), pp. 57-70

[6] A.B.J. Kuijlaars; K. McLaughlin Riemann–Hilbert analysis for Laguerre polynomials with large negative parameters, Comput. Meth. Funct. Theory, Volume 1 (2001), pp. 205-233

[7] X. Li; R. Wong On the asymptotics of the Meixner–Pollaczek polynomials and their zeros, Constructive Approximation, Volume 17 (2001), pp. 59-90

[8] J. Meixner Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion, J. London Math. Soc., Volume 9 (1934), pp. 6-13

[9] F.W.J. Olver Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders, J. Res. Nat. Bur. Standards Sect. B, Volume 63B (1959), pp. 131-169

[10] F. Pollaczek Sur une famille de polynômes orthogonaux qui contient les polynômes dʼHermite et de Laguerre comme cas limites, C. R. Acad. Sci. Paris, Ser. I, Volume 230 (1950), pp. 1563-1565

[11] W.Y. Qiu; R. Wong Asymptotic expansions for Riemann–Hilbert problems, Anal. Appl., Volume 6 (2008), pp. 269-298

[12] R. Wong; W.J. Zhang Uniform asymptotics for Jacobi polynomials with varying large negative parameters – a Riemann–Hilbert approach, Trans. Amer. Math. Soc., Volume 358 (2006), pp. 2663-2694

Cité par Sources :

Commentaires - Politique