On 𝐵𝐶 type basic hypergeometric orthogonal polynomials

Stokman, Jasper

doi:10.1090/S0002-9947-99-02551-9

On $BC$ type basic hypergeometric orthogonal polynomials
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Trans. Amer. Math. Soc. 352 (2000), 1527-1579 Request permission

Abstract:

The five parameter family of Koornwinder’s multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable $q$-Racah polynomials and multivariable big and little $q$-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known $q$-analogues of the Selberg integral are obtained.

References

George E. Andrews and Richard Askey, Enumeration of partitions: the role of Eulerian series and $q$-orthogonal polynomials, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 31, Reidel, Dordrecht-Boston, Mass., 1977, pp. 3–26. MR 519776
Jacques Labelle, Tableau d’Askey, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. xxxvi–xxxvii (French). MR 838967
K. Aomoto, On Elliptic Product Formulas for Jackson Integrals associated with Reduced Root Systems, J. of Alg. Comb. 8 (1998), pp. 115-126.
Richard Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), no. 6, 938–951. MR 595822, DOI 10.1137/0511084
Richard Askey and James Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols, SIAM J. Math. Anal. 10 (1979), no. 5, 1008–1016. MR 541097, DOI 10.1137/0510092
Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
T.H. Baker, P.J. Forrester, Multivariable Al-Salam & Carlitz polynomials associated with the type A $q$-Dunkl kernel, preprint (1997).
Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI 10.2307/2118632
Ivan Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), no. 1, 119–145. MR 1354956, DOI 10.1007/BF01231441
Ivan Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices 10 (1995), 483–515. MR 1358032, DOI 10.1155/S1073792895000341
Ivan Cherednik, Intertwining operators of double affine Hecke algebras, Selecta Math. (N.S.) 3 (1997), no. 4, 459–495. MR 1613515, DOI 10.1007/s000290050017
J. F. van Diejen, Self-dual Koornwinder-Macdonald polynomials, Invent. Math. 126 (1996), no. 2, 319–339. MR 1411136, DOI 10.1007/s002220050102
Jan F. van Diejen, On certain multiple Bailey, Rogers and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), no. 3, 483–508. MR 1474700, DOI 10.2977/prims/1195145326
J. F. van Diejen and J. V. Stokman, Multivariable $q$-Racah polynomials, Duke Math. J. 91 (1998), no. 1, 89–136. MR 1487981, DOI 10.1215/S0012-7094-98-09106-2
Ronald J. Evans, Multidimensional beta and gamma integrals, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 341–357. MR 1284073, DOI 10.1090/conm/166/01631
Robert A. Gustafson, Some $q$-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras, Trans. Amer. Math. Soc. 341 (1994), no. 1, 69–119. MR 1139492, DOI 10.1090/S0002-9947-1994-1139492-3
Robert A. Gustafson, A generalization of Selberg’s beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 97–105. MR 1001607, DOI 10.1090/S0273-0979-1990-15852-5
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
Laurent Habsieger, Une $q$-intégrale de Selberg et Askey, SIAM J. Math. Anal. 19 (1988), no. 6, 1475–1489 (French, with English summary). MR 965268, DOI 10.1137/0519111
Masahiko Ito, On a theta product formula for Jackson integrals associated with root systems of rank two, J. Math. Anal. Appl. 216 (1997), no. 1, 122–163. MR 1487257, DOI 10.1006/jmaa.1997.5665
Kevin W. J. Kadell, A proof of some $q$-analogues of Selberg’s integral for $k=1$, SIAM J. Math. Anal. 19 (1988), no. 4, 944–968. MR 946654, DOI 10.1137/0519066
Jyoichi Kaneko, $q$-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 5, 583–637. MR 1399617
R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Report 94-05 (Delft University of Technology, 1994).
Tom H. Koornwinder, Askey-Wilson polynomials for root systems of type $BC$, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 189–204. MR 1199128, DOI 10.1090/conm/138/1199128
I.G. Macdonald, Orthogonal polynomials associated with root systems, preprint (1988).
I.G. Macdonald, Some conjectures for Koornwinder’s orthogonal polynomials, unpublished manuscript (1988).
B. A. Bondarenko, Recurrence equations and arithmetic triangles, Voprosy Vychisl. i Prikl. Mat. 96 (1993), 137–172, 176–177 (Russian, with Russian and Uzbek summaries). MR 1352669
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), Exp. No. 797, 4, 189–207. Séminaire Bourbaki, Vol. 1994/95. MR 1423624
Masatoshi Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), no. 1, 16–77. MR 1413836, DOI 10.1006/aima.1996.0066
M. Noumi, Macdonald-Koornwinder polynomials and affine Hecke algebras, RIMS Kokyuroku 919 (1995), pp. 44–55 (in Japanese).
Masatoshi Noumi and Tetsuya Sugitani, Quantum symmetric spaces and related $q$-orthogonal polynomials, Group theoretical methods in physics (Toyonaka, 1994) World Sci. Publ., River Edge, NJ, 1995, pp. 28–40. MR 1413733
S. Sahi, Nonsymmetric Koornwinder polynomials and duality, preprint (1997).
Jasper V. Stokman, Multivariable big and little $q$-Jacobi polynomials, SIAM J. Math. Anal. 28 (1997), no. 2, 452–480. MR 1434045, DOI 10.1137/S0036141095287192
Jasper V. Stokman, Two limit transitions involving multivariable BC type Askey-Wilson polynomials, Quantum groups and quantum spaces (Warsaw, 1995) Banach Center Publ., vol. 40, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 415–428. MR 1481764
J.V. Stokman, Multivariable BC type Askey-Wilson polynomials with partly discrete orthogonality measure, Ramanujan J. 1 (1997), pp. 275–297.
Jasper V. Stokman and Tom H. Koornwinder, Limit transitions for BC type multivariable orthogonal polynomials, Canad. J. Math. 49 (1997), no. 2, 373–404. MR 1447497, DOI 10.4153/CJM-1997-019-9
J.V. Stokman, T.H. Koornwinder, On some limit cases of Askey-Wilson polynomials, J. Approx. Theory 95 (1998), pp. 310–330.
V. Tarasov, A. Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque No. 246 (1997).