Location of Repository

## On orthogonal polynomials and related discrete integrable systems

### Abstract

Orthogonal polynomials arise in many areas of mathematics and have been the subject\ud of interest by many mathematicians. In recent years this interest has often arisen from\ud outside the orthogonal polynomial community after their connection with integrable\ud systems was found. This thesis is concerned with the different ways these connections\ud can occur. We approach the problem from both perspectives, by looking for integrable\ud structures in orthogonal polynomials and by using an integrable structure to relate\ud different classes of orthogonal polynomials.\ud In Chapter 2, we focus on certain classes of semi-classical orthogonal polynomials. For\ud the classical orthogonal polynomials, the recurrence relations and differential equations\ud are well known and easy to calculate explicitly using an orthogonality relation or\ud generating function. However with semi-classical orthogonal polynomials, the recurrence\ud coefficients can no longer be expressed in an explicit form, but instead obeys systems\ud of non-linear difference equations. These systems are derived by deriving compatibility\ud relations between the recurrence relation and the differential equation. The compatibility\ud problem can be approached in two ways; the first is the direct approach using the\ud orthogonality relation, while the second introduces the Laguerre method, which derives\ud a differential system for semi-classical orthogonal polynomials. We consider some semiclassical\ud Hermite and Laguerre weights using the Laguerre method, before applying both\ud methods to a semi-classical Jacobi weight. While some of the systems derived will have\ud been seen before, most of them (at least not to our knowledge) have not been acquired\ud from this approach.\ud Chapter 3 considers a singular integral transform that is related to the Gel’fand-Levitan\ud equation, which provides the inverse part of the inverse scattering method (a solution\ud method of integrable systems). These singular integral transforms constitute a dressing\ud method between elementary (bare) solutions of an integrable system to more complicated\ud solutions of the same system. In the context of this thesis we are interested in adapting this method to the case of polynomial solutions and study dressing transforms between\ud different families of polynomials, in particular between certain classical orthogonal\ud polynomials and their semi-classical deformations.\ud In chapter 4, a new class of orthogonal polynomials are considered from a formal\ud approach: a family of two-variable orthogonal polynomials related through an elliptic\ud curve. The formal approach means we are interested in those relations that can be derived,\ud without specifying a weight function. Thus, we are mainly concerned with recursive\ud structures, particularly on their explicit derivation so that a series of elliptic polynomials\ud can be constructed. Using generalized Sylvester identities, recurrence relations are\ud derived and we consider the consistency of their coefficients and the compatibility\ud between the two relations. Although the chapter focuses on the structure of the recurrence\ud relations, some applications are also presented

Publisher: Applied Mathematics (Leeds)
Year: 2006
OAI identifier: oai:etheses.whiterose.ac.uk:101

### Citations

1. (1990). 2D Gravity from Matrix Models, (Invited Lecture at Johns Hopkins Workshop,
2. (1990). A direct proof of the Christoffel-Darboux identity and its equivalence to the recurrence relationship
3. (1973). A discrete version of the inverse scattering problem
4. (2002). A generalisation of the Chebyshev polynomials,
5. (1985). A Pad´ e-type approach to non-classical orthgonal polynomials,
6. (1984). A proof of Freud’s conjecture about orthogonal polynomials related to jxj exp(¡x2m) for integer m,
7. (1992). A uniﬁed approach to various orthogonalities,
8. Abelian Functions, Abel’s theorem and the allied theory of theta functions,
9. (1977). Acc´ el´ eration de la Convergence en Analyse Num´ erique,
10. (1978). An introduction to orthogonal polynomials,
11. (1997). an Toda-type differential equations for the recurrence coefﬁcients of orthogonal polynomials and Freud transformation,
12. Ausf¨ uhrung der ganzzahligen Multiplikation der elliptischen Funktionen, Journal f¨ ur die reine u.
13. (1990). Basic Hypergeometric Series, Encycl. of Math. and its Appls. Vol 35,
14. (1990). Boundary Value Problems,
15. (1965). Characterization of certain classes of orthogonal polynomials related to elliptic functions, Annali di Matematica pura ed applicata 67
16. (2005). Classical and Quantum Orthogonal Polynomials
17. Classiﬁcation of integrable equations on quad-graphs. The consistency approach,
18. (1947). Complete collected works , 2 ,
19. (1980). Direct Methods in Soliton Theory,
20. (1999). Discrete conformal maps and surfaces, in:
21. (1948). Einf¨ urung in die Determinantentheorie,
22. (1990). Elements of the Theory of Elliptic Curves, AMS Publ.
23. (2000). Elliptic Polynomials, (Chapman and Hall CRC,
24. (1975). Essentials of Pad´ e Approximants,
25. (1990). Exactly solvable ﬁelds theories of closed strings,
26. (1995). Formal orthogonality on an algebraic curve,
27. (1936). Formal properties of orthogonal polynomials in two variables,
28. (1999). Freud equations for orthogonal polynomials as discrete Painlev´ e equations in:
29. (1993). From Continuous to Discrete Painlev´
30. (1973). Higher transcendental functions: Bessel functions, functions of a parabolic cylinder, orthogonal polynomials, 2nd edition; Russian translation,
31. (1979). Holomorphic Fiberings and Nonlinear Equations,
32. (2002). Integrable systems on quad-graphs,
33. (1968). Integrals of nonlinear equations of evolution and solitary waves,
34. (1978). Inverse Scattering, Orthogonal Polynomials, and Linear Estimation, Topics in Func.
35. (1926). Kamp´ e F´ eriet, Fonctions hyperg´ eometriques et hyperspheriques, Polynˆ omes d’Hermite, Gauthier-Villars,
36. (1974). Korteweg-deVries equation and generalizations. VI. Methods for exact solution,
37. Kuijlaars Multiple orthogonal polynomials of a mixed type and non-intersecting brownian motions math.CA/0511470.
38. (1994). Laguerre-Freuds equations for the recurrence coefﬁcients of semi-classical orthogonal polynomials,
39. (1878). M´ emoire sur l’approximation des fonctions de tr´ es grands nombres,
40. (1996). Malaschonok Various Proofs of Sylvester’s (Determinant)
41. (1967). Method for solving the Korteweg-de Vries equation,
42. MOPS: Multivariate Orthogonal Polynomials (symbolically)
43. (1998). Multiple Orthogonal Polynomials
44. (1980). Nonlinear Wave Equations and Constrained Harmonic Motion,
45. (1990). Nonperturbative Two-Dimensional Quantum Gravity,
46. (1998). On a family of orthogonal polynomials related to elliptic functions,
47. (1975). On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices,
48. (1936). On derivatives of orthogonal polynomials I,
49. (1974). On some two-dimensional analogues of classical orthogonal polynomials, Latv mat. ezhegodnik
50. (1976). On the coefﬁcients in the recursion formulae of Orthogonal Polynomials,
51. (1951). On the determination of a differential equation from its spectral function,
52. (1956). Ordinary Differential Equations,
53. (1999). Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Aproach,
54. (1975). Orthogonal Polynomials II
55. (1974). Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators,
56. (1967). Orthogonal polynomials in two variables,
57. (2001). Orthogonal Polynomials of Several Variables, Encycl. of Math. and its Appls. Vol 81,
58. Orthogonal Polynomials, in:
59. (1969). Orthogonale Polynome, Birkh¨ auser,
60. (1974). OrthogonalPolynomialsform the viewpointof scattering theory
61. (1980). Pad´ e-Type Approximation and General Orthogonal Polynomials,
62. (1995). Painlev´ e-type differential equations for the recurrence coefﬁcients of semi-classical polynomials,
63. (2005). Pi˜ nar Weak classical orthogonal polynomials in two variables,
64. (1983). Polynˆ omes Orthogonaux Formels, Applications,
65. (1999). Proofs and Conﬁrmations: The Story of the Alternating Sign Matrix Conjecture,
66. (1985). Semi-classical orthogonal polynomials,
67. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering,
68. (2001). Some Orthogonal Polynomials Related to Elliptic Functions,
69. (1999). Special Functions Encylopedia of Mathematics and its Application’s 71
70. (1961). Statistical Theory of the Energy Levels of Complex Systems I,
71. (1961). Statistical Theory of the Energy Levels of Complex Systems II,
72. (1961). Statistical Theory of the Energy Levels of Complex Systems III,
73. (1963). Statistical Theory of the Energy Levels of Complex Systems IV,
74. (1963). Statistical Theory of the Energy Levels of Complex Systems V,
75. Sur l’interpolation,
76. Sur la r´ eduction en fractions continues d’une fonction qui satisfait ` a une ´ equationlin´ eairedupremierordre ` acoefﬁcientsrationnelsBulletindelaS.M.F.,
77. (1972). Sur la r´ eduction en fractions continues d’une fraction qui satisfait ` a une ´ equation diff´ erentielle lin´ eaire du premier ordre dont les coefﬁcients sont rationnels
78. Sur les fractions continues,
79. Sur les valeurs limites des integrales,
80. (1930). Sur une classe ´ etendue de fractions continues alg´ ebraiques et sur les polynomes de Tchebycheff correspondants,
81. (1968). Sylvesters identity and multistep integer preserving Gaussian elimination,
82. (1998). Symmetric Functions and Hall Polynomials,
83. The Askey-scheme of hypergeometric orthogonal polynomials and its q analogue math.CA/9602214
84. The Gelfand-Levitan method, in the vector valued case,
85. (1974). The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems,
86. (1972). The Pad´ e table and it’s relation to certain algorithm of numerical analysis,
87. (1955). Theory of Ordinary Differential Equations,
88. (1983). Transformation group for soliton equations,
89. (1975). Two-variable analogues of the classical orthogonal polynomials, in: Theory and Application of Special Functions,
90. Uber die Gaussische Quadratur und eine Verallgemeinerung derselben,
91. (1935). Uber die Jacobischen Polynome und zwei verwandte Polynomklassen,
92. (1989). unbaum, The Kadomtsev-Petviashvili Equation: An Alternative Approach to “Rank Two”
93. Vries On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves,
94. Witte Bi-orthogonal polynomials on the Unit Circle, Regular Semi-Classical Weights and Integrable Systems,
95. Witte Discrete Painlev´ e Equations for a class of PVI ¿-functions given as U(N) averages,
96. Zur Theorie der elliptischen Functionen, Journal f¨ ur die reine u. angewandte

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.