Introduction to Leonard pairs

Journal of Computational and Applied Mathematics

Volumn 153, Issue 1-2, 2003, Pages 463-475

  Terwilliger, Paul ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Askey scheme; Askey Wilson polynomial; Leonard pair; q Racah polynomial; Tridiagonal pair

Indexed keywords

MATHEMATICAL TRANSFORMATIONS; POLYNOMIALS; VECTORS;

LEONARD PAIRS;

COMPUTATIONAL METHODS;

MATHEMATICAL ANALYSIS;

EID: 0037382602     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(02)00600-3     Document Type: Article

References (42)

1. C. Ahn, K. Shigemoto, Onsager algebra and integrable lattice models, Modern Phys. Lett. A 6 (38) (1991) 3509-3515.
2. G. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
3. E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, London, 1984.
4. A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer, Berlin, 1989.
5. J.S. Caughman IV, The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete Math. 196 (1999) 65-95.
6. B. Curtin, Distance-regular graphs which support a spin model are thin, in: Proceedings of the 16th British Combinatorial Conference, London, 1997, Discrete Math. 197/198 (1999) 205-216.
7. B. Curtin, K. Nomura, Distance-regular graphs related to the quantum enveloping algebra of sl(2), J. Algebraic Combin. 12 (2000) 25-36.
8. E. Date, S.S. Roan, The structure of quotients of the Onsager algebra by closed ideals, J. Phys. A 33 (2000) 3275-3296.
9. B. Davies, Onsager's algebra and the Dolan-Grady condition in the non-self-dual case, J. Math. Phys. 32 (1991) 2945-2950.
10. L. Dolan, M. Grady, Conserved charges from self-duality, Phys. Rev. D (3) 25 (1982) 1587-1604.
11. G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990.
12. Ya.I. Granovskiǐ, I.M., Lutzenko, A.S. Zhedanov, Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Phys. 217 (1992) 1-20.
13. Ya.A. Granovskiǐ, A.S. Zhedanov, Nature of the symmetry group of the 6j-symbol. Zh. 94 (1988) 49-54.
14. q(2), J. Phys. A 25 (1992) L1029-L1032.
15. q(2), J. Phys. A 26 (1993) L357-L359.
16. Ya.I. Granovskiǐ, A.S. Zhedanov, Spherical q-functions, J. Phys. A 26 (1993) 4331-4338.
17. F.A. Grünbaum, Some bispectral musings, in: The Bispectral Problem, Montreal, PQ, 1997, American Mathematical Society Providence, RI, 1998, pp. 31-45.
18. F.A. Grünbaum, L. Haine, The q-version of a theorem of Bochner, J. Comput. Appl. Math. 68 (1996) 103-114.
19. F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: an extension of the Krall polynomials, Internat. Math. Res. Notices 8 (1997) 359-392.
20. F.A. Grünbaum, L. Haine, Some functions that generalize the Askey-Wilson polynomials, Comm. Math. Phys. 184 (1997) 173-202.
21. F.A. Grünbaum, L. Haine, On a q-analogue of the string equation and a generalization of the classical orthogonal polynomials, in: Algebraic Methods and q-Special Functions, Montréal, QC, 1996, American Mathematical Society, Providence, RI, 1999, pp. 171-181.
22. F.A. Grünbaum, L. Haine, The Wilson bispectral involution: some elementary examples, in: Symmetries and Integrability of Difference Equations, Canterbury, 1996, Cambridge University Press, Cambridge, 1999, pp. 353-369.
23. F.A. Grünbaum L. Haine, Horozov, E. Some functions that generalize the Krall-Laguerre polynomials, J. Comp. Appl. Math. 106 (1999) 271-297.
24. T. Ito, K. Tanabe, P. Terwilliger, Some algebra related to P- and Q-polynomial association schemes, in: Codes and Association Schemes, Piscataway, NJ, 1999, American Mathematical Society, Providence, RI, 2001, pp. 167-192.
25. C. Kassel, Quantum Groups, Springer, New York, 1995.
26. R. Koekoek, R.F. Swarttouw, The Askey scheme of hypergeometric orthogonal polyomials and its q-analog, Report 98-17, Delft University of Technology, The Netherlands, 1998, Available at http://aw.twi.tudelft.nl/~koekoek/research.html.
27. H.T. Koelink, Askey-Wilson polynomials and the quantum su(2) group: survey and applications, Acta Appl. Math. 44 (1996) 295-352.
28. H.T. Koelink, q-Krawtchouk polynomials as spherical functions on the Hecke algebra of type B, Trans. Amer. Math. Soc. 352 (2000) 4789-4813.
29. H.T. Koelink, J. Van der Jeugt, Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. Math. Anal. 29 (1998) 794-822.
30. H.T. Koelink, J. Van der Jeugt, Bilinear generating functions for orthogonal polynomials, Constr. Approx. 15 (1999) 481-497.
31. T.H. Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the su(2) quantum group, SIAM J. Math. Anal. 24 (1993) 795-813.
32. D. Leonard, Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal. 13 (1982) 656-663.
33. H. Rosengren, Multivariable Orthogonal Polynomials as Coupling Coefficients for Lie and Quantum Algebra Representations, Centre for Mathematical Sciences, Lund University, Sweden, 1999.
34. P. Terwilliger, The incidence algebra of a uniform poset, Math. Appl. 20 (1990) 193-212.
35. P. Terwilliger, The subconstituent algebra of an association scheme, J. Algebraic Combin. 1 (1992) 363-388.
36. P. Terwilliger, Introduction to Leonard pairs and Leonard systems, Sūrikaisekikenkyūsho Kōkyūroku 1109 (1999) 67-79 (Algebraic combinatorics, Kyoto, 1999.)
37. P. Terwilliger, Leonard pairs from 24 points of view, in: Proceedings of the Special Functions 2000, Tempe AZ, 2000, accepted for publication.
38. P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001) 149-203.
39. P. Terwilliger, Two relations that generalize the q-Serre relations and the Dolan-Grady relations, in: Physics and Combinatorics 1999, Nagoya, World Scientific, River Edge, NJ, 2001, pp. 377-398.
40. A.S. Zhedanov, Hidden symmetry of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991) 190-204.
41. q(2) algebra: Cartesian version and overlaps, Modern Phys. Lett. A 7 (1992) 1589-1593.
42. A.S. Zhedanov, Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials, J. Phys. A 26 (1993) 4633-4641.