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q-Selberg Integrals and Macdonald Polynomials

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Abstract

Using the theory of Macdonald polynomials, a number of q-integrals of Selberg type are proved.

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References

  1. G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.

  2. K. Aomoto, “Connection formulas of the q-analog de Rham cohomology, in Functional Analysis on the Eve of the 21st Century,” (S. Gindikin et al., eds.), Prog. in Math. 131, Birkhauser, Boston, MA, 1995, vol. 1, pp. 1–12.

  3. R. Askey, “Some basic hypergeometric extensions of integrals of Selberg and Andrews,” SIAM J. Math. Anal. 11 (1980), 938–951.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Askey, “Ramanujan's extensions of the gamma and beta functions,” Amer. Math. Monthly 87 (1980), 346–359.

    MATH  MathSciNet  Google Scholar 

  5. R. Askey, “Beta integrals in Ramanujan's papers, his unpublished work and further examples,” in Ramanujan Revisited (G.E. Andrews et al. eds.), Academic Press, Boston, MA, 1988, pp. 561–590.

  6. T.H. Baker and P.J. Forrester, “Transformation formulas for multivariable basic hypergeometric series,” Methods Appl. Anal. 6 (1999), 147–164.

    Google Scholar 

  7. R.J. Evans, “Multidimensional beta and gamma integrals,” in The Rademacher Legacy to Mathematics (G.E. Andrews et al., eds.), Contemp. Math. 166, AMS, Providence, RI, 1994, pp. 341–357.

  8. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990.

  9. L. Habsieger, “Une q-intégrale de Selberg et Askey,” SIAM J. Math. Anal. 19 (1988), 1475–1489.

    Article  MATH  MathSciNet  Google Scholar 

  10. L.K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Translations of Mathematical Monographs, vol. 6, AMS, Providence, RI, 1979.

  11. K.W.J. Kadell, “A proof of some q-analogues of Selberg's integral for k = 1,” SIAM J. Math. Anal. 19 (1988), 944–968.

    MATH  MathSciNet  Google Scholar 

  12. K.W.J. Kadell, “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.

    MATH  MathSciNet  Google Scholar 

  13. K.W.J. Kadell, “An integral for the product of two Selberg–Jack symmetric polynomials,” Compositio Math. 87 (1993), 5–43.

    MATH  MathSciNet  Google Scholar 

  14. K.W.J. Kadell, “The Selberg–Jack symmetric functions,” Adv. Math. 130 (1997), 33–102.

    Article  MATH  MathSciNet  Google Scholar 

  15. K.W.J. Kadell, “A simple proof of an Aomoto-type extension of Askey's last conjectured Selberg q-integral,” J. Math. Anal. Appl. 261 (2001), 419–440.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Kaneko, “q-Selberg integrals and Macdonald polynomials,” Ann. Sci. École Norm. Sup. (4) 29 (1996), 583–637.

  17. J. Kaneko, “Constant term identities of Forrester–Zeilberger–Cooper,” Discrete Math. 173 (1997), 79–90.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Kaneko, “A 1Ψ1 summation theorem for Macdonald polynomials,” Ramanujan J. 2 (1998), 379–386.

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Krattenthaler, “Schur function identities and the number of perfect matchings of holey Aztec rectangles,” in q-Series from a Contemporary Perspective (M.E.H. Ismail and D.W. Stanton, eds.), Contemp. Math. 254, AMS, Providence, RI, 2000, pp. 335–349.

  20. I.G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, New-York, 1995.

  21. I.G. Macdonald, Hypergeometric Series II, unpublished manuscript.

  22. S.C. Milne, “Summation theorems for basic hypergeometric series of Schur function argument,” in Progress in Approximation Theory, (A.A. Gonchar and E.B. Saff eds.), Springer Ser. Comput. Math. 19, Springer, New York, 1992, pp. 51–77.

  23. W.G. Morris, “Constant Term Identities for Finite Affine Root Systems: Conjectures and Theorems,” Ph.D. dissertation, University of Wisconsin–Madison, 1982.

  24. A. Selberg, “Bemerkninger om et multipelt integral,” Norske Mat. Tidsskr. 26 (1944), 71–78.

    MATH  MathSciNet  Google Scholar 

  25. V. Tarasov and A. Varchenko, “Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups,” Astérisque 246 (SMF, Paris, 1997).

  26. Z. Yan, “A class of generalized hypergeometric functions in several variables,” Canad. J. Math. 44 (1992), 1317–1338.

    MATH  MathSciNet  Google Scholar 

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Correspondence to S. Ole Warnaar.

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2000 Mathematics Subject Classification: Primary—33D05, 33D52, 33D60

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Warnaar, S.O. q-Selberg Integrals and Macdonald Polynomials. Ramanujan J 10, 237–268 (2005). https://doi.org/10.1007/s11139-005-4849-7

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  • DOI: https://doi.org/10.1007/s11139-005-4849-7

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