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Plancherel–Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems

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Abstract

We study the Plancherel–Rotach asymptotics of four families of orthogonal polynomials: the Chen–Ismail polynomials, the Berg–Letessier–Valent polynomials, and the Conrad–Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.

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References

  1. Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyed, Edinburgh (1965). English translation

    MATH  Google Scholar 

  2. Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  3. Berg, C., Christensen, J.P.R.: Density questions in the classical theory of moments. Ann. Inst. Fourier (Grenoble) 31, 99–114 (1981). French summary

    Article  MATH  MathSciNet  Google Scholar 

  4. Berg, C., Pedersen, H.L.: On the order and type of the entire functions associated with an indeterminate Hamburger moment problem. Ark. Mat. 32, 1–11 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berg, C., Valent, G.: The Nevanlinna parameterization for some indeterminate Stieltjes moment problems associated with birth and death processes. Methods Appl. Anal. 1, 169–209 (1994)

    MATH  MathSciNet  Google Scholar 

  6. Cao, L.H., Li, Y.T.: Linear difference equations with a transition point at the origin. Anal. Appl. (2013). doi:10.1142/S0219530513500371

    Google Scholar 

  7. Chen, Y., Ismail, M.E.H.: Some indeterminate moment problems and Freud-like weights. Constr. Approx. 14, 439–458 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gilewicz, J., Leopold, E., Valent, G.: New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes. J. Comput. Appl. Math. 178, 235–245 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hethcote, H.W.: Error bounds for asymptotic approximations of zeros of transcendental functions. SIAM J. Math. Anal. 1, 147–152 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2009), paperback edition

    MATH  Google Scholar 

  12. Ismail, M.E.H., Li, X.: Bounds for extreme zeros of orthogonal polynomials. Proc. Am. Math. Soc. 115, 131–140 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ismail, M.E.H., Li, X.: Plancherel–Rotach asymptotics for q-orthogonal polynomials. Constr. Approx. 37, 341–356 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ismail, M.E.H., Masson, D.R.: q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals. Trans. Am. Math. Soc. 346, 63–116 (1994)

    MATH  MathSciNet  Google Scholar 

  15. Kriecherbauer, T., McLaughlin, K.T.-R.: Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Not. 6, 299–333 (1999)

    Article  MathSciNet  Google Scholar 

  16. Kuijlaars, A.B.J., Van Assche, W.: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99(1), 167–197 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lee, K.F., Wong, R.: Uniform asymptotic expansions of the Tricomi–Carlitz polynomials. Proc. Am. Math. Soc. 138(7), 2513–2519 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Letessier, J., Valent, G.: Some exact solutions of the Kolmogorov boundary value problem. J. Approx. Theory Appl. 4, 97–117 (1988)

    MATH  MathSciNet  Google Scholar 

  19. Levin, A.L., Lubinsky, D.S.: Orthogonal polynomials and Christoffel functions for exp(−|X|α), α≤1. J. Approx. Theory 80(2), 219–252 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Levin, E., Lubinsky, D.S.: Orthogonal Polynomials for Exponential Weights. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 4. Springer, New York (2001)

    Book  MATH  Google Scholar 

  21. Levin, E., Lubinsky, D.S.: Orthogonal polynomials for weights close to indeterminacy. J. Approx. Theory 147(2), 129–168 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Shohat, J., Tamarkin, J.D.: The Problem of Moments. American Mathematical Society, Providence (1950), revised edition

    MATH  Google Scholar 

  23. Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)

    Google Scholar 

  24. Van Assche, W., Geronimo, J.S.: Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients. Rocky Mt. J. Math. 19(1), 39–49 (1989)

    Article  MATH  Google Scholar 

  25. Van Fossen Conrad, E.: Some continued fraction expansions of Laplace transforms of elliptic functions. Doctoral Dissertation, The Ohio State University, Columbus, Ohio (2002)

  26. Van Fossen Conrad, E., Flajolet, P.: The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion. Sémin. Lothar. Comb. 54 (2006). 44 pages

  27. Vanlessen, M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. Constr. Approx. 25, 125–175 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Wang, X.-S., Wong, R.: Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. 10, 215–235 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang, Z., Wong, R.: Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94(1), 147–194 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wang, Z., Wong, R.: Linear difference equations with transition points. Math. Comput. 74(250), 629–653 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank Professor Doron Lubinsky for his helpful comments and discussions. We are also grateful to the editors and referees for their valuable suggestions and extremely careful reading of the manuscript. Dan Dai was partially supported by a grant from City University of Hong Kong (Project No. 7002883) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 100910). Mourad E.H. Ismail was partially supported by NPST Program of King Saud University (Project No. 10-MAT1293-02) and by the DSFP at King Saud University in Riyadh, and by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1014111).

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Hong Kong, Hong Kong SAR

    Dan Dai

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Mourad E. H. Ismail

  3. King Saud University, Riyadh, Saudi Arabia

    Mourad E. H. Ismail

  4. Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO, 63701, USA

    Xiang-Sheng Wang

Authors
  1. Dan Dai
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  2. Mourad E. H. Ismail
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  3. Xiang-Sheng Wang
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Correspondence to Xiang-Sheng Wang.

Additional information

Communicated by Tom H. Koornwinder.

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Dai, D., Ismail, M.E.H. & Wang, XS. Plancherel–Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems. Constr Approx 40, 61–104 (2014). https://doi.org/10.1007/s00365-013-9215-1

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  • DOI: https://doi.org/10.1007/s00365-013-9215-1

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