Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 014, 35 pages      arXiv:2106.03421      https://doi.org/10.3842/SIGMA.2022.014

$q$-Selberg Integrals and Koornwinder Polynomials

Jyoichi Kaneko
Department of Mathematical Sciences, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan

Received June 23, 2021, in final form February 14, 2022; Published online February 28, 2022

Abstract
We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.

Key words: Koornwinder polynomials; quadratic norm formula; antisymmetrization; $q$-Selberg integral; Mehta's integral.

pdf (543 kb)   tex (36 kb)  

References

  1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  2. Aomoto K., On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), 115-126.
  3. Askey R., Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938-951.
  4. Askey R., Richards D., Selberg's second beta integral and an integral of Mehta, in Probability, Statistics, and Mathematics, Academic Press, Boston, MA, 1989, 27-39.
  5. Baker T.H., Dunkl C.F., Forrester P.J., Polynomial eigenfunctions of the Calogero-Sutherland-Moser models with exchange terms, in Calogero-Moser-Sutherland Models (Montréal, QC, 1997), CRM Ser. Math. Phys., Springer, New York, 2000, 37-51.
  6. Baker T.H., Forrester P.J., Generalizations of the $q$-Morris constant term identity, J. Combin. Theory Ser. A 81 (1998), 69-87.
  7. Baratta W., Some properties of Macdonald polynomials with prescribed symmetry, Kyushu J. Math. 64 (2010), 323-343, arXiv:1001.3134.
  8. Belbachir H., Boussicault A., Luque J.G., Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde, J. Algebra 320 (2008), 3911-3925, arXiv:0709.3021.
  9. Di Francesco P., Gaudin M., Itzykson C., Lesage F., Laughlin's wave functions, Coulomb gases and expansions of the discriminant, Internat. J. Modern Phys. A 9 (1994), 4257-4351, arXiv:hep-th/9401163.
  10. Forrester P.J., Warnaar S.O., The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 489-534, arXiv:0710.3981.
  11. Gessel I.M., Lv L., Xin G., Zhou Y., A unified elementary approach to the Dyson, Morris, Aomoto, and Forrester constant term identities, J. Combin. Theory Ser. A 115 (2008), 1417-1435, arXiv:0701066.
  12. Gustafson R.A., A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97-105.
  13. Habsieger L., Une $q$-intégrale de Selberg et Askey, SIAM J. Math. Anal. 19 (1988), 1475-1489.
  14. Hamada S., Proof of Baker-Forrester's constant term conjecture for the cases $N_1=2,3$, Kyushu J. Math. 56 (2002), 243-266.
  15. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  16. Ito M., Forrester P.J., A bilateral extension of the $q$-Selberg integral, Trans. Amer. Math. Soc. 369 (2017), 2843-2878, arXiv:1309.0001.
  17. Kadell K.W.J., A proof of Askey's conjectured $q$-analogue of Selberg's integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), 969-986.
  18. Kaneko J., $q$-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), 583-637.
  19. Kaneko J., On Baker-Forrester's constant term conjecture, J. Ramanujan Math. Soc. 18 (2003), 349-367.
  20. Károlyi G., Nagy Z.L., Petrov F.V., Volkov V., A new approach to constant term identities and Selberg-type integrals, Adv. Math. 277 (2015), 252-282, arXiv:1312.6369.
  21. Koornwinder T.H., Askey-Wilson polynomials for root systems of type $BC$, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204.
  22. Lusztig G., Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635.
  23. Macdonald I.G., The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161-174.
  24. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  25. Mehta M.L., Dyson F.J., Statistical theory of the energy levels of complex systems. V, J. Math. Phys. 4 (1963), 713-719.
  26. Morris II W.G., Constant term identities for finite and affine root systems, conjectures and theorems, Ph.D. Thesis, The University of Wisconsin, Madison, 1982.
  27. Noumi M., Macdonald-Koornwinder polynomials and affine Hecke rings, Sūrikaisekikenkyūsho Kōkyūroku 919 (1995), 44-55.
  28. Sahi S., Some properties of Koornwinder polynomials, in $q$-Series from a Contemporary Perspective (South Hadley, MA, 1998), Contemp. Math., Vol. 254, Amer. Math. Soc., Providence, RI, 2000, 395-411.
  29. Selberg A., Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71-78.
  30. Stokman J.V., On $BC$ type basic hypergeometric orthogonal polynomials, Trans. Amer. Math. Soc. 352 (2000), 1527-1579, arXiv:q-alg/9707005.
  31. Stokman J.V., Koornwinder polynomials and affine Hecke algebras, Int. Math. Res. Not. 2000 (2000), 1005-1042, arXiv:math.QA/0002090.
  32. Stokman J.V., Lecture notes on Koornwinder polynomials, in Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004, 145-207.
  33. Stokman J.V., Koornwinder T.H., Limit transitions for BC type multivariable orthogonal polynomials, Canad. J. Math. 49 (1997), 373-404.
  34. Warnaar S.O., $q$-Selberg integrals and Macdonald polynomials, Ramanujan J. 10 (2005), 237-268.
  35. Warnaar S.O., The ${\mathfrak{sl}}_3$ Selberg integral, Adv. Math. 224 (2010), 499-524, arXiv:0901.4176.
  36. Xin G., Zhou Y., A Laurent series proof of the Habsieger-Kadell $q$-Morris identity, Electron. J. Combin. 21 (2014), 3.38, 16 pages, arXiv:1302.6642.

Previous article  Next article  Contents of Volume 18 (2022)