Abstract
In this paper we present an addition to Askey’s scheme of q- hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function. The generalized orthogonality relations and the second order q-differenee equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are discussed shortly.
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References
G. E. Andrews and R. Askey, Classical orthogonal polynomials, in: “Polynômes Orthogonaux et Applications”, Lecture Notes in Mathematics, Vol. 1171, Springer-Verlag, 1985, pp. 36–62.
R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54, no. 319 (1985).
F. Bonechi, N. Ciccoli, R. Giachetti, E. Sorace, and M. Tarlini, Free q-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group, Comm. Math. Phys. 175 (1996), 161–176.
B. M. Brown, W. D. Evans and M. E. H. Ismail, The Askey-Wilson polynomials and q-Sturm-Liouville problems, Math. Proc. Cambridge Philos. Soc. 119 (1996), 1–16.
J. Bustoz and S. K. Suslov, Basic analog of Fourier series on a q-quadratic grid, Methods Appl. Anal. 5 (1998), 1–38.
N. Ciccoli, E. Koelink and T. H. Koornwinder, q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations, Methods Appl. Anal. 6 (1999), 109–127.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, 1990.
F. A. Grünbaum and L. Haine, Some functions that generalize the Askey-Wilson polynomials, Comm. Math. Phys. 184 (1997), 173–202.
D. P. Gupta, M. E. H. Ismail and D. R. Masson, Contiguous relations, basic hypergeometric functions, and orthogonal polynomials III. Associated continuous dual q-Hahn polynomials, J. Comput. Appl. Math. 68 (1996), 115–149.
M. E. H. Ismail, The zeros of basic Bessel functions, the function J v+ax(x), and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1–19.
M. E. H. Ismail, D. R. Masson and S. K. Suslov, The q-Bessel function on a q-quadratic grid, in: “Algebraic Methods and q-Special Functions”, J. F. van Diejen and L. Vinet, eds., CRM Proc. Lect. Notes, Vol. 22, Amer. Math. Soc, 1999, pp. 183–200.
M. E. H. Ismail and M. Rahman, The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), 201–237.
T. Kakehi, Eigenfunction expansion associated with the Casimir operator on the quantum group SU q(1,1), Duke Math. J. 80 (1995), 535–573.
T. Kakehi, T. Masuda and K. Ueno, Spectral analysis of a q-difference operator which arises from the quantum SU(1,1) group, J. Operator Theory 33 (1995), 159–196.
R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology Report no. 98–17, 1998.
H. T. Koelink, On quantum groups and q-special functions, dissertation, Rijksuniversiteit Leiden, 1991.
H. T. Koelink, The quantum group of plane motions and the Hahn-Exton q-Bessel function, Duke Math. J. 76 (1994), 483–508.
H. T. Koelink, The quantum group of plane motions and basic Bessel functions, Indag. Math. (N. S.) 6 (1995), 197–211.
H. T. Koelink, Yet another basic analogue of Graf’s addition formula, J. Comput. Appl. Math. 68 (1996), 209–220.
H. T. Koelink, Askey-Wilson polynomials and the quantum SU(2) group: survey and applications, Acta Appl. Math. 44 (1996), 295–352.
H. T. Koelink, Some basic Lommel polynomials, J. Approx. Theory 96 (1999), 345–365.
E. Koelink and J. V. Stokman, The big q-Jacobi function transform, preprint 40 p., math.CA/9904111, 1999.
E. Koelink and J. V. Stokman, with an appendix by M. Rahman, Fourier transforms on the quantum SU(1,1) group, preprint 77 p., math.QA/9911163, 1999.
E. Koelink and J. V. Stokman, The Askey-Wilson function transform, preprint 19 p., math.CA/0004053, 2000.
T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in: “Special Functions: Group Theoretical Aspects and Applications”, R. A. Askey, T. H. Koornwinder and W. Schempp, eds., Reidel, 1984, pp. 1–85.
T. H. Koornwinder, Group theoretic interpretations of Askey’s scheme of hypergeometric orthogonal polynomials, in: “Orthogonal Polynomials and their Applications”, Lecture Notes in Mathematics, Vol. 1329, Springer, 1988, pp. 46–72.
T. H. Koornwinder, Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group, SIAM J. Math. Anal. 24 (1993), 795–813.
T. H. Koornwinder, Compact quantum groups and q-special functions, in: “Representations of Lie Groups and Quantum Groups”, Pitman Res. Notes Math. Ser., Vol. 311, Longman Sci. Tech., 1994, pp. 46–128.
T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), 445–461.
T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Saburi and K. Ueno, Uni-tary representations of the quantum group SUq(l,l): Structure of the dual space of Uq(sl(2)), Lett. Math. Phys. 19 (1990), 197–194; II: Matrix elements of unitary representations and the basic hypergeometric functions, 195-204.
D. S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl. 81 (1981), 20–47.
M. Noumi, Quantum groups and q-orthogonal polynomials. Towards a realization of Askey-Wilson polynomials on SUq(2), in: “Special Functions”, M. Kashiwara and T. Miwa, eds., ICM-90 Satellite Conference Proceedings, Springer-Verlag, 1991, pp. 260–288.
M. Noumi and K. Mimachi, Askey-Wilson polynomials as spherical functions on SUq(2), “Quantum Groups,” LNM, vol. 1510, Springer Verlag, 1992, 98–103.
H. Rosengren, A new quantum algebraic interpretation of the Askey-Wilson polynomials, Contemp. Math. 254 (2000), 371–394.
S. N. M. Ruijsenaars, A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type, Comm. Math. Phys. 206 (1999), 639–690.
J. V. Stokman and T. H. Koornwinder, On some limit cases of Askey-Wilson polynomials, J. Approx. Theory 95 (1998), 310–330.
S. K. Suslov, Some orthogonal very well poised 8ø7-functions, J. Phys. A 30 (1997), 5877–5885.
S. K. Suslov, Some orthogonal very-well-poised 8ϕ7-functions that generalize Askey-Wilson polynomials, MSRI preprint, 1997; to appear in The Ramanujan Journal.
L. L. Vaksman and L. I. Korogodskii, The algebra of bounded functions on the quantum group of motions of the plane and q-analogues of Bessel functions, Soviet Math. Dokl. 39 (1989), 173–177.
L. L. Vaksman and L. I. Korogodskii, Spherical functions on the quantum group SU(1,1) and a q-analogue of the Mehler-Fock formula, Funct. Anal. Appl. 25 (1991), 48–49.
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1944.
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Koelink, E., Stokman, J.V. (2001). The Askey-Wilson Function Transform Scheme. In: Bustoz, J., Ismail, M.E.H., Suslov, S.K. (eds) Special Functions 2000: Current Perspective and Future Directions. NATO Science Series, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0818-1_9
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DOI: https://doi.org/10.1007/978-94-010-0818-1_9
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