Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-11T04:30:19.166Z Has data issue: false hasContentIssue false

4 - Orthogonal Polynomials, their Recursions, and Functional Equations

Published online by Cambridge University Press:  05 July 2011

By
Mourad E. H. Ismail
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Mourad E. H. Ismail
Affiliation:
University of Central Florida
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
Get access

Summary

Abstract

These lectures contain a brief introduction to orthogonal polynomials with some applications to nonlinear recurrence relation. We discuss the spectral theory of orthogonal polynomials. We also mention other applications which may not be well known to the integrable systems community. For example we treat birth and death processes, the J-matrix method, and inversions of large Hankel matrices. The spectral theory of the Askey–Wilson operators on bounded and unbounded intervals is briefly mentioned.

Introduction

Many mathematicians and physicists think orthogonal polynomials are polynomial solutions of Sturm–Liouville differential equations. This is usually emphasized in textbooks written with an applied flavor. Although the Jacobi, Hermite, Laguerre polynomials and their special cases are indeed solutions to such equations, all orthogonal polynomials arise from difference equations. In addition to satisfying second order difference equations in the degree they may also satisfy an operator equation in a continuous variable.

In this article we include a brief account of certain aspects of orthogonal polynomials which may appeal to specialists in difference equations. In particular we give many instances in the theory of orthogonal polynomials where difference equations play a fundamental role. We also discuss the spectral theory of orthogonal polynomials and review the direct and inverse problems for orthogonal polynomials, which is the theory of linear symmetric second order difference equations. We also show how structure relations for orthogonal polynomials lead to nonlinear difference equations satisfied by the recurrence coefficients of the orthogonal polynomials under very special initial conditions.

Type
Chapter
Information
Symmetries and Integrability of Difference Equations , pp. 115 - 138
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Akhiezer, N. I. 1965. The Classical Moment Problem and Some Related Questions in Analysis. New York: Hafner.Google Scholar
[2] Alhaidari, A. D. 2004. Exact L2 series solution of the Dirac–Coulomb problem for all energies. Ann. Physics, 312(1), 144–160.CrossRefGoogle Scholar
[3] Askey, R., and Ismail, M. 1984. Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc., 49(300).Google Scholar
[4] Askey, R., and Wilson, J. 1985. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319).Google Scholar
[5] Bank, E., and Ismail, M. E. H. 1985. The attractive Coulomb potential polynomials. Constr. Approx., 1(2), 103–119.CrossRefGoogle Scholar
[6] Bauldry, W. C. 1990. Estimates of asymptotic Freud polynomials on the real line. J. Approx. Theory, 63(2), 225–237.CrossRefGoogle Scholar
[7] Biedenharn, L. C., and Louck, J. D. 1981. The Racah–Wigner Algebra in Quantum Theory. Encyclopedia Math. Appl., vol. 9. Reading, MA: Addison-Wesley.Google Scholar
[8] Bonan, S. S., and Clark, D. S. 1990. Estimates of the Hermite and Freud polynomials. J. Approx. Theory, 63(2), 210–224.CrossRefGoogle Scholar
[9] Brown, B. M., and Christiansen, J. S. 2005. On the Krein and Friedrichs extensions of a positive Jacobi operator. Expo. Math., 23(2), 179–186.CrossRefGoogle Scholar
[10] Brown, B. M., Evans, W. D., and Ismail, M. E. H. 1996. The Askey–Wilson polynomials and q-Sturm–Liouville problems. Math. Proc. Cambridge Philos. Soc., 119(1), 1–16.CrossRefGoogle Scholar
[11] Chen, Y., and Ismail, M. E. H. 1997. Ladder operators and differential equations for orthogonal polynomials. J. Phys. A, 30(22), 7818–7829.CrossRefGoogle Scholar
[12] Chen, Y., and Ismail, M. E. H. 2008. Ladder operators for q-orthogonal polynomials. J. Math. Anal. Appl., 345(1), 1–10.CrossRefGoogle Scholar
[13] Choi, M. D. 1983. Tricks or treats with the Hilbert matrix. Amer. Math. Monthly, 90(5), 301–312.CrossRefGoogle Scholar
[14] Christiansen, J. S., and Ismail, M. E. H. 2006. A moment problem and a family of integral evaluations. Trans. Amer. Math. Soc., 358(9), 4071–4097.CrossRefGoogle Scholar
[15] Christiansen, J. S., and Koelink, E. 2006. Self-adjoint difference operators and classical solutions to the Stieltjes–Wigert moment problem. J. Approx. Theory, 140(1), 1–26.CrossRefGoogle Scholar
[16] Favard, J. 1935. Sur les polynômes de Tchebicheff. C. R. Acad. Sci. Paris, 200, 2052–2053.Google Scholar
[17] Freud, G. 1976. On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A, 76(1), 1–6.Google Scholar
[18] Gasper, G., and Rahman, M. 2004. Basic Hypergeometric Series. 2nd edn. Encyclopedia Math. Appl., vol. 96. Cambridge: Cambridge Univ. Press.CrossRefGoogle Scholar
[19] Grammaticos, B., and Ramani, A. 2004. Discrete Painlevé equations: a review. Pages 245–321 of: Discrete Integrable Systems. Lecture Notes in Math., vol. 644. Berlin: Springer.CrossRefGoogle Scholar
[20] Heller, E. J., Reinhardt, W. P., and Yamani, H. A. 1973. On an “equivalent quadrature” calculation of matrix elements of (z - p2/2m) using an L2-expansion technique. J. Comput. Phys., 13(4), 536–549.CrossRefGoogle Scholar
[21] Ismail, M. E. H. 1993. Ladder operators for q-1-Hermite polynomials. C. R. Math. Rep. Acad. Sci. Canada, 15(6), 261–266.Google Scholar
[22] Ismail, M. E. H. 2003. Difference equations and quantized discriminants for q-orthogonal polynomials. Adv. in Appl. Math., 30(3), 562–589.CrossRefGoogle Scholar
[23] Ismail, M. E. H. 2005a. Asymptotics of q-orthogonal polynomials and a q-Airy function. Int. Math. Res. Not., 2005(18), 1063–1088.CrossRefGoogle Scholar
[24] Ismail, M. E. H. 2005b. Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia Math. Appl., vol. 98. Cambridge: Cambridge Univ. Press.CrossRefGoogle Scholar
[25] Ismail, M. E. H., and Koelink, E.The J-matrix method. Adv. Appl. Math to appear.
[26] Ismail, M. E. H., and Mansour, Z. S.q-analogues of Freud weights and nonlinear difference equations.
[27] Ismail, M. E. H., and Simeonov, P. 2009. q-difference operators for orthogonal polynomials. J. Comput. Appl. Math., 233(3), 749–761.CrossRefGoogle Scholar
[28] Ismail, M. E. H., Johnston, S., and Mansour, Z. S.Structure relations or q-polynomials and some applications. Applicable Anal. to appear.
[29] Ismail, M. E. H., Nikolova, I., and Simeonov, P. 2004. Difference equations and discriminants for discrete orthogonal polynomials. Ramanujan J., 8(4), 475–502.CrossRefGoogle Scholar
[30] Karlin, S., and McGregor, J. 1957a. The classification of birth and death processes. Trans. Amer. Math. Soc., 86, 366–400.CrossRefGoogle Scholar
[31] Karlin, S., and McGregor, J. 1957b. The differential equations of birth-and-death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc., 85, 489–546.CrossRefGoogle Scholar
[32] Koekoek, R., and R. F., Swarttouw. 1998. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Reports of the Faculty of Technical Mathematics and Informatics 98–17. Delft University of Technology, Delft.Google Scholar
[33] Lubinsky, D. S., Mhaskar, H. N., and Saff, E. B. 1988. A proof of Freud's conjecture for exponential weights. Constr. Approx., 4(1), 65–83.CrossRefGoogle Scholar
[34] Mhaskar, H. N. 1990. Bounds for certain Freud polynomials. J. Approx. Theory, 63(2), 238–254.CrossRefGoogle Scholar
[35] Munger, C. T. 2007. Ideal basis sets for the Dirac Coulomb problem: eigenvalue bounds and convergence proofs. J. Math. Phys., 48(2), 022301.CrossRefGoogle Scholar
[36] Rakhmanov, E. A. 1982. Asymptotic properties of orthogonal polynomials on the real axis. Mat. Sb. (N. S.), 119(161)(2), 163–203. in Russian.Google Scholar
[37] Shohat, J. A., and Tamarkin, J. D. 1950. The Problem of Moments. Math. Surveys, vol. 1. Providence, RI: Amer. Math. Soc.Google Scholar
[38] Simon, B. 1998. The classical moment as a self-adjoint finite difference operator. Adv. Math., 137(1), 82–203.CrossRefGoogle Scholar
[39] Stone, M. H. 1932. Linear Transformations in Hilbert Space and their Application to Analysis. New York: Amer. Math. Soc.CrossRefGoogle Scholar
[40] Szegő, G. 1975. Orthogonal Polynomials. 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Providence, RI: Amer. Math. Soc.Google Scholar
[41] Van Assche, W. 2007. Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials. Pages 687–725 of: Difference Equations, Special Functions and Orthogonal Polynomials. Hackensack, NJ: World Sci. Publ.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×