Book contents
- Frontmatter
- Contents
- List of figures
- List of contributors
- Preface
- Introduction
- 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
- 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete
- 3 Definitions and Predictions of Integrability for Difference Equations
- 4 Orthogonal Polynomials, their Recursions, and Functional Equations
- 5 Discrete Painlevé Equations and Orthogonal Polynomials
- 6 Generalized Lie Symmetries for Difference Equations
- 7 Four Lectures on Discrete Systems
- 8 Lectures on Moving Frames
- 9 Lattices of Compact Semisimple Lie Groups
- 10 Lectures on Discrete Differential Geometry
- 11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
- References
4 - Orthogonal Polynomials, their Recursions, and Functional Equations
Published online by Cambridge University Press: 05 July 2011
Book contents
- Frontmatter
- Contents
- List of figures
- List of contributors
- Preface
- Introduction
- 1 Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
- 2 Painlevé Equations: Continuous, Discrete and Ultradiscrete
- 3 Definitions and Predictions of Integrability for Difference Equations
- 4 Orthogonal Polynomials, their Recursions, and Functional Equations
- 5 Discrete Painlevé Equations and Orthogonal Polynomials
- 6 Generalized Lie Symmetries for Difference Equations
- 7 Four Lectures on Discrete Systems
- 8 Lectures on Moving Frames
- 9 Lattices of Compact Semisimple Lie Groups
- 10 Lectures on Discrete Differential Geometry
- 11 Symmetry Preserving Discretization of Differential Equations and Lie Point Symmetries of Differential-Difference Equations
- References
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 - 138Publisher: Cambridge University PressPrint publication year: 2011
References
[1] 1965. The Classical Moment Problem and Some Related Questions in Analysis. New York: Hafner.Google Scholar
[2] 2004. Exact L2 series solution of the Dirac–Coulomb problem for all energies. Ann. Physics, 312(1), 144–160.CrossRefGoogle Scholar
[3] , and 1984. Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc., 49(300).Google Scholar
[4] , and 1985. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319).Google Scholar
[5] , and 1985. The attractive Coulomb potential polynomials. Constr. Approx., 1(2), 103–119.CrossRefGoogle Scholar
[6] 1990. Estimates of asymptotic Freud polynomials on the real line. J. Approx. Theory, 63(2), 225–237.CrossRefGoogle Scholar
[7] , and 1981. The Racah–Wigner Algebra in Quantum Theory. Encyclopedia Math. Appl., vol. 9. Reading, MA: Addison-Wesley.Google Scholar
[8] , and 1990. Estimates of the Hermite and Freud polynomials. J. Approx. Theory, 63(2), 210–224.CrossRefGoogle Scholar
[9] , and 2005. On the Krein and Friedrichs extensions of a positive Jacobi operator. Expo. Math., 23(2), 179–186.CrossRefGoogle Scholar
[10] , , and 1996. The Askey–Wilson polynomials and q-Sturm–Liouville problems. Math. Proc. Cambridge Philos. Soc., 119(1), 1–16.CrossRefGoogle Scholar
[11] , and 1997. Ladder operators and differential equations for orthogonal polynomials. J. Phys. A, 30(22), 7818–7829.CrossRefGoogle Scholar
[12] , and 2008. Ladder operators for q-orthogonal polynomials. J. Math. Anal. Appl., 345(1), 1–10.CrossRefGoogle Scholar
[13] 1983. Tricks or treats with the Hilbert matrix. Amer. Math. Monthly, 90(5), 301–312.CrossRefGoogle Scholar
[14] , and 2006. A moment problem and a family of integral evaluations. Trans. Amer. Math. Soc., 358(9), 4071–4097.CrossRefGoogle Scholar
[15] , and 2006. Self-adjoint difference operators and classical solutions to the Stieltjes–Wigert moment problem. J. Approx. Theory, 140(1), 1–26.CrossRefGoogle Scholar
[17] 1976. On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A, 76(1), 1–6.Google Scholar
[18] , and 2004. Basic Hypergeometric Series. 2nd edn. Encyclopedia Math. Appl., vol. 96. Cambridge: Cambridge Univ. Press.CrossRefGoogle Scholar
[19] , and 2004. Discrete Painlevé equations: a review. Pages 245–321 of: Discrete Integrable Systems. Lecture Notes in Math., vol. 644. Berlin: Springer.CrossRefGoogle Scholar
[20] , , and 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] 1993. Ladder operators for q-1-Hermite polynomials. C. R. Math. Rep. Acad. Sci. Canada, 15(6), 261–266.Google Scholar
[22] 2003. Difference equations and quantized discriminants for q-orthogonal polynomials. Adv. in Appl. Math., 30(3), 562–589.CrossRefGoogle Scholar
[23] 2005a. Asymptotics of q-orthogonal polynomials and a q-Airy function. Int. Math. Res. Not., 2005(18), 1063–1088.CrossRefGoogle Scholar
[24] 2005b. Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia Math. Appl., vol. 98. Cambridge: Cambridge Univ. Press.CrossRefGoogle Scholar
[25] , and The J-matrix method. Adv. Appl. Math to appear.
[26] , and q-analogues of Freud weights and nonlinear difference equations.
[27] , and 2009. q-difference operators for orthogonal polynomials. J. Comput. Appl. Math., 233(3), 749–761.CrossRefGoogle Scholar
[28] , , and Structure relations or q-polynomials and some applications. Applicable Anal. to appear.
[29] , , and 2004. Difference equations and discriminants for discrete orthogonal polynomials. Ramanujan J., 8(4), 475–502.CrossRefGoogle Scholar
[30] , and 1957a. The classification of birth and death processes. Trans. Amer. Math. Soc., 86, 366–400.CrossRefGoogle Scholar
[31] , and 1957b. The differential equations of birth-and-death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc., 85, 489–546.CrossRefGoogle Scholar
[32] , and . 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] , , and 1988. A proof of Freud's conjecture for exponential weights. Constr. Approx., 4(1), 65–83.CrossRefGoogle Scholar
[34] 1990. Bounds for certain Freud polynomials. J. Approx. Theory, 63(2), 238–254.CrossRefGoogle Scholar
[35] 2007. Ideal basis sets for the Dirac Coulomb problem: eigenvalue bounds and convergence proofs. J. Math. Phys., 48(2), 022301.CrossRefGoogle Scholar
[36] 1982. Asymptotic properties of orthogonal polynomials on the real axis. Mat. Sb. (N. S.), 119(161)(2), 163–203. in Russian.Google Scholar
[37] , and 1950. The Problem of Moments. Math. Surveys, vol. 1. Providence, RI: Amer. Math. Soc.Google Scholar
[38] 1998. The classical moment as a self-adjoint finite difference operator. Adv. Math., 137(1), 82–203.CrossRefGoogle Scholar
[39] 1932. Linear Transformations in Hilbert Space and their Application to Analysis. New York: Amer. Math. Soc.CrossRefGoogle Scholar
[40] 1975. Orthogonal Polynomials. 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Providence, RI: Amer. Math. Soc.Google Scholar
[41] 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
- 1
- Cited by