Convolutions for orthogonal polynomials from lie and quantum algebra representations

SIAM Journal on Mathematical Analysis

Volumn 29, Issue 3, 1998, Pages 794-822

  Koelink, H T ^a   Van Der Jeugt, J ^b  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

^b GHENT UNIVERSITY   (Belgium)

Author keywords

Convolution; Lie algebra; Orthogonal polynomials; Quantum algebra

Indexed keywords

EID: 0032383984     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003614109630673X     Document Type: Article

References (36)

1. W. A. AL-SALAM. Characterization theorems for orthogonal polynomials, in Orthogonal Polynomials: Theory and Practice, P. Nevai, ed., NATO ASI series C 294, Kluwer, 1990, pp. 1-24.
2. W. A. AL-SALAM AND T. S. CHIHARA, Convolutions of orthonormal polynomials, SIAM J. Math. Anal., 7 (1976), pp. 16-28.
3. R. ASKEY, Continuous Hahn polynomials, J. Math. Phys. A: Math. Gen., 18 (1985), pp. L1017-L1019.
4. R. ASKEY AND J. WILSON, A set of orthogonal polynomials that generalize the Racah coefficients or 6 - j symbols, SIAM J. Math. Anal., 10 (1979), pp. 1008-1016.
5. R. ASKEY AND J. WILSON, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), pp. 1-55.
6. N. M. ATAKISHIYEV AND S. K. SUSLOV, The Hahn and Meixner polynomials of imaginary argument and some of their applications, J. Math. Phys. A: Math. Gen., 18 (1985), pp. 1583-1596.
7. J. M. BEREZANSKIǐ, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr. 17, Amer. Math. Soc., 1968.
8. 1 , 1), J. Phys. A: Math. Gen., 26 (1993), pp. 2139-2151.
9. V. CHARI AND A. PRESSLEY, A Guide to Quantum Groups, Cambridge University Press, Cambridge, UK, 1994.
10. C. F. DUNKL, Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type, SIAM J. Alg. Disc. Meth., 1 (1980), pp. 137-151.
11. C. F. DUNKL, A difference equation and Hahn polynomials in two variables, Pacific J. Math., 92 (1981), pp. 57-71.
12. C. F. DUNKL, Orthogonal polynomials with symmetry of order three, Canad. J. Math., 36 (1984), pp. 685-717.
13. A. ERDÉLYI, W. MAGNUS, F. OBERHETTINGER, AND F. G. TRICOMI, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York. 1953.
14. G. GASPER AND M. RAHMAN. Basic Hypergeometric Series, Cambridge University Press, Cambridge, UK, 1990.
15. q(2), J. Phys A: Math. Gen., 25 (1992), pp. L1029-L1032.
16. Y. I. GRANOVSKII AND A. S. ZHEDANOV, New construction of 3nj-symbols, J. Phys A: Math. Gen., 26 (1993), pp. 4339-4344.
17. q(2), J. Group Theory Phys., 1 (1993), pp. 161-171.
18. V. A. GROZA AND I. I. KACHURIK, Addition and product theorems for Krawtchouk, Hahn and Racah q-polynomials, Dokl. Akad. Nauk Ukraine SSR Ser. A. 89 (1990), pp. 3-6 (in Russian).
19. E. G. KALNINS, H. L. MANOCHA, AND W. MILLER JR., Models of q-algebra representations: Tensor products of special unitary and oscillator algebras, J. Math. Phys., 33 (1992), pp. 2365-2383.
20. 2), q-orthogonal polynomials and invariants of links, in Infinite-dimensional Lie Algebras and Groups, V.G. Kac, ed., World Scientific, Singapore, 1989, pp. 285-339.
21. A. U. KLIMYK AND I. I. KACHURIK, Spectra, eigenvectors and overlap functions for representation operators of q-deformed algebras, Comm. Math. Phys., 175 (1996), pp. 89-111.
22. R. KOEKOEK AND R. F. SWARTTOUW, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-analogue. Report 94-05, Technical University Delft, Delft, the Netherlands 1994; also available online from ftp.twi.tudelft.nl in directory /pub/publications/tech-reports.
23. H. T. KOELINK, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996), pp. 887-898.
24. H. T. KOELINK, Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications, Acta Appl. Math., 44 (1996), pp. 295-352.
25. T. H. KOORNWINDER, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys., 30 (1989), pp. 767-769.
26. T. H. KOORNWINDER, Positive convolution structures associated with quantum groups, in Probability Measures on Groups X, H. Heyer, ed., Plenum Press, New York, 1991, pp. 249-268.
27. T. H. KOORNWINDER, Askey-Wilson polynomials for root systems of type BC, Contemp. Math., 138 (1992), pp. 189-204.
28. T. H. KOORNWINDER, Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group, SIAM J. Math. Anal., 24 (1993), pp. 795-813.
29. 2(ℤ) and the su(1, 1) Lie algebra, SIAM J. Math. Anal., 22 (1991), pp. 1133-1146.
30. q(2), in Special Functions, M. Kashiwara and T. Miwa, eds., ICM-90 Satellite Conference Proceedings, Springer, New York, 1991, pp. 260-288.
31. q(2), Proc. Japan Acad. Ser. A, 66 (1990), pp. 146-149.
32. q(2), in Quantum Groups, LNM 1510, P. P. Kulish, ed., Springer, New York, 1992, pp. 98-103.
33. G. SZEGO, Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, RI, 1975.
34. J. VAN DER JEUGT, Coupling coefficients for Lie algebra representations and addition formulas for special functions, J. Math. Phys., 38 (1997), pp. 2728-2740.
35. N. J. VILENKIN AND A. U. KLIMYK, Representation of Lie Groups and Special Functions, Vols. 1-3, Kluwer Academic Publishers, Norwell, MA, 1991-3.
36. J. A. WILSON, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal., 11 (1980), pp. 690-701.