A noncommutative version of the bispectral problem

Journal of Computational and Applied Mathematics

Volumn 161, Issue 1, 2003, Pages 99-118

  Grunbaum F ^a   Iliev, Plamen ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL METHODS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA;

BISPECTRAL PROBLEMS;

PROBLEM SOLVING;

MATHEMATICAL ANALYSIS;

EID: 0242541922     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(03)00548-X     Document Type: Article

References (49)

1. Adler M. Moser J. On a class of polynomials connected with the Korteweg-de Vries equation Comm. Math. Phys. 61 1978 1-30
2. Airault H. McKean H.P. Moser J. Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem Comm. Pure Appl. Math. 30 1977 95-148
3. Bakalov B. Horozov E. Yakimov M. Bispectral algebras of commuting ordinary differential operators Comm. Math. Phys. 190 1997 331-373
4. Y. Berest, Huygens principle and the bispectral problem, in: J. Harnad, A. Kasman (Eds.), The Bispectral Problem, (Montreal, PQ, 1997), CRM Proceeding Lecture Notes, Vol. 14, American Mathematical Society, Providence, RI, 1998, pp. 11-30.
5. Berest Y. Kasman A. D-modules and Darboux transformations Lett. Math. Phys. 43 1998 279-294
6. Y. Berest, A.P. Veselov, The Huygens principle and integrability, (Russian) Uspekhi Mat. Nauk 49 (1994) 7-78; translation in Russian Math. Surveys 49 (1994) 5-77.
7. Berest Y. Wilson G. Classification of rings of differential operators on affine curves Internat. Math. Res. Notices 2 1999 105-109
8. Bochner S. Über Sturm-Liouvillesche Polynomsysteme Math. Z. 29 1929 730-736
9. M. Cantero, L. Moral, L. Velazquez, Differential properties of matrix orthogonal polynomials, preprint math. CA/0205094.
10. Chalykh O.A. Feigin M.V. Veselov A.P. Multidimensional Baker-Akhiezer functions and Huygens' principle Comm. Math. Phys. 206 1999 533-566
11. Duistermaat J.J. Grünbaum F.A. Differential equations in the spectral parameter Comm. Math. Phys. 103 1986 177-240
12. A. Duran, Matrix inner product having a matrix symmetric second order differential operators, Rocky Mountain J. Math. 27 (2) (1997) 585-600.
13. Felder G. Markov Y. Tarasov V. Varchenko A. Differential equations compatible with KZ equations Math. Phys. Anal. Geom. 3 2000 139-177
14. Gangolli R. Varadarajan V.S. Harmonic analysis of spherical functions on real reductive groups Series title: Ergebnisse der Mathematik und ihrer Grenzgebeite Vol. 101 1988 Springer Berlin, New York
15. Granovskii Ya.I. Lutzenko I.M. Zhedanov A.S. Mutual integrability, quadratic algebras, and dynamical symmetry Ann. Phys. 217 1992 1-20
16. Gekhtman M.I. Non-abelian nonlinear lattice equations on finite interval J. Phys. A 26 1993 6303-6317
17. F.A. Grünbaum, Some bispectral musings, in: The Bispectral Problem (Montreal, PQ, 1997), CRM Proceeding Lecture Notes, Vol. 14, American Mathematical Society, Providence, RI, 1998, pp. 11-30.
18. F.A. Grünbaum, The bispectral problem: an overview, in: J. Bustoz et al. (Eds.), Special Functions 2000: Current Perspective and Future Directions, 2001, pp. 129-140.
19. Grünbaum F.A. Matrix valued Jacobi polynomials Bull. Sci. Math. 127 3 2003 207-214
20. Grünbaum F.A. Haine L. A theorem of Bochner revisited Fokas A.S. Gelfand I.M. Algebraic Aspects of Integrable Systems Progressing Nonlinear Differential Equations Vol. 26 1997 143-172 Birkhäuser Boston
21. Grünbaum F.A. Haine L. Associated polynomials, spectral matrices and the Bispectral problem Meth. Appl. Anal. 6 6 2000 209-224
22. Grünbaum F.A. Pacharoni I. Tirao J. A matrix valued solution to Bochner's problem J. Physics A: Math. Gen. 34 2001 10 647-10 656
23. Grünbaum F.A. Pacharoni I. Tirao J.A. Matrix valued spherical functions associated to the complex projective plane J. Funct. Anal. 188 2002 350-441
24. Grünbaum F.A. Pacharoni I. Tirao J.A. Matrix valued spherical functions associated to the three dimensional hyperbolic space Internat. J. Math. 13 7 2002 727-784
25. 2(ℂ), math.RT/0202304.
26. F.A. Grünbaum, I. Pacharoni, J.A. Tirao, Matrix valued orthogonal polynomials of the Jacobi type, submitted.
27. L. Haine, Beyond the classical orthogonal polynomials (Montreal PQ, 1997), CRM Proceeding Lecture Notes, Vol. 14, American Mathematical Society, Providence, 1998, pp. 47-65.
28. Haine L. The Bochner-Krall problem: some new perspectives Bustoz J. Ismail M.E.H. Suslov S.K. Special Functions 2000: Current Perspective and Future Directions NATO Sciences Series II. Mathematics, Physics and Chemistry Vol. 30 2001 141-178 Kluwer Academic Publishers Dordrecht
29. Haine L. Iliev P. Commutative rings of difference operators and an adelic flag manifold Internat. Math. Res. Notices 6 2000 281-323
30. Haine L. Iliev P. A rational analogue of the Krall polynomials, Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000) J. Phys. A 34 2001 2445-2457
31. J. Harnad, A. Kasman (Eds.), The Bispectral Problem, CRM Proceedings and Lectures notes. Vol. 14, American Mathematical Society, Providence, 1998.
32. Horozov E. Kasman A. Darboux transformations of bispectral quantum integrable systems Lett. Math. Phys. 49 1999 131-143
33. Jodar L. Company R. Navarro E. Laguerre matrix polynomials and systems of second order differential equations Appl. Numer. Math. 15 1994 53-63
34. Jodar L. Legua M. Solving second order matrix differential equations with regular singular points avoiding the increase of the problem dimension Appl. Math. Comput. 53 1993 191-206
35. Krall H.L. Certain differential equations for the Tchebychev polynomials Duke Math. J. 4 1938 705-718
36. I.M. Krichever, Algebraic curves and non-linear difference equations (Russian), Uspekhi Mat. Nauk 33 (1978) 215-216; translation in Russian Math. Surveys 33 (1978) 255-256.
37. Mumford D. An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations Nagata M. Proceedings of International Symposium on Algebraic Geometry, Kyoto 1977 1978 115-153 Kinokuniya Book Store Tokyo
38. Nijhoff F. Chalykh O. Bispectral rings of difference operators Russian Math. Surveys 54 1999 644-645
39. I. Pacharoni, J.A. Tirao, Three term recursion relation for spherical functions associated to the complex projective plane, to appear.
40. Ruijsenaars S.N.M. Special functions defined by analytic difference equations Bustoz J. Ismail M.E.H. Suslov S.K. Special Functions 2000: Current Perspective and Future Directions NATO Sciences Series II. Mathematics, Physics and Chemistry Vol. 30 2001 281-333 Kluwer Academic Publishers Dordrecht
41. Spiridonov V.P. Zhedanov A.S. Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids Bustoz J. Ismail M.E.H. Suslov S.K. Special Functions 2000: Current Perspective and Future Directions NATO Sciences Series II. Mathematics, Physics and Chemistry Vol. 30 2001 365-388 Kluwer Academic Publishers Dordrecht
42. Tirao J.A. Spherical Functions Rev. de la Unión Matem. Argentina 28 1977 75-98
43. van Diejen J.F. Kirillov A.N. Formulas for q-spherical functions using inverse scattering theory of reflectionless Jacobi operators Comm. Math. Phys. 210 2 2000 335-369
44. van Diejen J.F. Vinet L. The quantum dynamics of the compactified trigonometric Ruijsenaars-Schneider model Comm. Math. Phys. 197 1 1998 33-74
45. van Moerbeke P. Mumford D. The spectrum of difference operators and algebraic curves Acta Math. 143 1979 93-154
46. Wilson G. Bispectral commutative differential operators J. Reine Angew. Math. 442 1993 177-204
47. Wilson G. Collisions of Calogero-Moser Particles and an Adelic Grassmanian, with an appendix by I.G. Macdonald Invent. Math. 133 1998 1-41
48. Zubelli J. Differential equations in the spectral parameter for matrix differential operators Physica D Nonlinear Phenomena 43 1990 269-287
49. Zubelli J. Rational solutions of nonlinear evolution equations, vertex operators and bispectrality J. Differential Equations 97 1992 71-98