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Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we study a family of orthogonal polynomials $\{\phi_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $\phi_n(z)$ as the polynomial degree $n$ tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials $\phi_n(z)$ is provided.
Keywords: uniform asymptotics; orthogonal polynomials; coherent states; three-term recurrence relation. 
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     author = {Dan Dai and Weiying Hu and Xiang-Sheng Wang},
     title = {Uniform {Asymptotics} of {Orthogonal} {Polynomials} {Arising} from {Coherent} {States}},
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     year = {2015},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a69/}
}
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Dan Dai; Weiying Hu; Xiang-Sheng Wang. Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a69/

[1] Ali S. T., Antoine J.-P., Gazeau J.-P., Coherent states, wavelets and their generalizations, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 2000 | DOI | MR | Zbl

[2] Ali S. T., Ismail M. E. H., “Some orthogonal polynomials arising from coherent states”, J. Phys. A: Math. Theor., 45 (2012), 125203, 16 pp., arXiv: 1111.7038 | DOI | MR | Zbl

[3] Aptekarev A. I., Kalyagin V. A., Saff E. B., “Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials”, Constr. Approx., 30 (2009), 175–223 | DOI | MR | Zbl

[4] Askey R., Wimp J., “Associated Laguerre and Hermite polynomials”, Proc. Roy. Soc. Edinburgh Sect. A, 96 (1984), 15–37 | DOI | MR | Zbl

[5] Cao L.-H., Li Y.-T., “Linear difference equations with a transition point at the origin”, Anal. Appl. (Singap.), 12 (2014), 75–106, arXiv: 1303.4846 | DOI | MR | Zbl

[6] Combescure M., Robert D., Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, Dordrecht, 2012 | DOI | MR | Zbl

[7] Coussement E., Coussement J., Van Assche W., “Asymptotic zero distribution for a class of multiple orthogonal polynomials”, Trans. Amer. Math. Soc., 360 (2008), 5571–5588, arXiv: math.CA/0606440 | DOI | MR | Zbl

[8] Dai D., Ismail M. E. H., Wang X.-S., “Plancherel–Rotach asymptotic expansion for some polynomials from indeterminate moment problems”, Constr. Approx., 40 (2014), 61–104, arXiv: 1302.6196 | DOI | MR | Zbl

[9] Deift P. A., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lecture Notes in Mathematics, 3, New York University, Courant Institute of Mathematical Sciences, New York, 1999 | MR

[10] Gazeau J.-P., Coherent states in quantum physics, Wiley-VCH, Weinheim, 2009 | DOI

[11] Geronimo J. S., WKB and turning point theory for second order difference equations: external fields and strong asymptotics for orthogonal polynomials, arXiv: 0905.1684

[12] Geronimo J. S., Smith D. T., “WKB (Liouville–Green) analysis of second order difference equations and applications”, J. Approx. Theory, 69 (1992), 269–301 | DOI | MR | Zbl

[13] Geronimo J. S., Smith D. T., Van Assche W., “Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients”, J. Approx. Theory, 72 (1993), 141–158 | DOI | MR | Zbl

[14] Glauber R. J., “The quantum theory of optical coherence”, Phys. Rev., 130 (1963), 2529–2539 | DOI | MR

[15] Glauber R. J., “Coherent and incoherent states of the radiation field”, Phys. Rev., 131 (1963), 2766–2788 | DOI | MR

[16] Klauder J. R., “Continuous-representation theory. I: Postulates of continuous-representation theory”, J. Math. Phys., 4 (1963), 1055–1058 | DOI | MR | Zbl

[17] Klauder J. R., “Continuous-representation theory. II: Generalized relation between quantum and classical dynamics”, J. Math. Phys., 4 (1963), 1058–1073 | DOI | MR | Zbl

[18] Kuijlaars A. B. J., Van Assche W., “The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients”, J. Approx. Theory, 99 (1999), 167–197 | DOI | MR | Zbl

[19] Máté A., Nevai P., Totik V., “Asymptotics for orthogonal polynomials defined by a recurrence relation”, Constr. Approx., 1 (1985), 231–248 | DOI | MR | Zbl

[20] Olver F. W. J., Asymptotics and special functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997 | MR | Zbl

[21] Olver F. W. J., Lozier D. W., Boisvert R. F., Clark C. W. (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010 | MR

[22] Schrödinger E., “Der stetige Übergang von der Mikro- zur Makromechanik”, Naturwissenschaften, 14 (1926), 664–666 | DOI | Zbl

[23] Sudarshan E. C. G., “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams”, Phys. Rev. Lett., 10 (1963), 277–279 | DOI | MR | Zbl

[24] Szegő G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975

[25] Van Assche W., Geronimo J. S., “Asymptotics for orthogonal polynomials on and off the essential spectrum”, J. Approx. Theory, 55 (1988), 220–231 | DOI | MR | Zbl

[26] Van Assche W., Geronimo J. S., “Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients”, Rocky Mountain J. Math., 19 (1989), 39–49 | DOI | MR | Zbl

[27] Wang X.-S., “Plancherel–Rotach asymptotics of second-order difference equations with linear coefficients”, J. Approx. Theory, 188 (2014), 1–18, arXiv: 1403.1281 | DOI | MR | Zbl

[28] Wang X.-S., Wong R., “Asymptotics of orthogonal polynomials via recurrence relations”, Anal. Appl. (Singap.), 10 (2012), 215–235, arXiv: 1101.4371 | DOI | MR | Zbl

[29] Wang Z., Wong R., “Asymptotic expansions for second-order linear difference equations with a turning point”, Numer. Math., 94 (2003), 147–194 | DOI | MR | Zbl

[30] Wang Z., Wong R., “Linear difference equations with transition points,”, Math. Comp., 74 (2005), 629–653 | DOI | MR | Zbl

[31] Wong R., Asymptotic approximations of integrals, Classics in Applied Mathematics, 34, SIAM, Philadelphia, PA, 2001 | DOI | MR | Zbl