Kaneko, Jyoichi \(q\)-Selberg integrals and Koornwinder polynomials. (English) Zbl 1506.33012 SIGMA, Symmetry Integrability Geom. Methods Appl. 18, Paper 014, 35 p. (2022). Summary: 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. MSC: 33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 05A30 \(q\)-calculus and related topics 11B65 Binomial coefficients; factorials; \(q\)-identities Keywords:Koornwinder polynomials; quadratic norm formula; antisymmetrization; \(q\)-Selberg integral; Mehta’s integral PDFBibTeX XMLCite \textit{J. Kaneko}, SIGMA, Symmetry Integrability Geom. Methods Appl. 18, Paper 014, 35 p. (2022; Zbl 1506.33012) Full Text: DOI arXiv References: [1] Andrews, George E. and Askey, Richard and Roy, Ranjan, Special functions, Encyclopedia of Mathematics and its Applications, 71, xvi+664, (1999), Cambridge University Press, Cambridge · Zbl 0920.33001 [2] Aomoto, Kazuhiko, On elliptic product formulas for {J}ackson integrals associated with reduced root systems, Journal of Algebraic Combinatorics. An International Journal, 8, 2, 115-126, (1998) · Zbl 0918.33013 [3] Askey, Richard, Some basic hypergeometric extensions of integrals of {S}elberg and {A}ndrews, SIAM Journal on Mathematical Analysis, 11, 6, 938-951, (1980) · Zbl 0458.33002 [4] Askey, Richard and Richards, Donald, Selberg’s second beta integral and an integral of {M}ehta, Probability, Statistics, and Mathematics, 27-39, (1989), Academic Press, Boston, MA · Zbl 0683.33001 [5] Baker, T. H. and Forrester, P. J., Generalizations of the {\(q\)}-{M}orris constant term identity, Journal of Combinatorial Theory. Series A, 81, 1, 69-87, (1998) · Zbl 0890.05006 [6] Baker, T. H. and Dunkl, C. F. and Forrester, P. J., Polynomial eigenfunctions of the {C}alogero–{S}utherland–{M}oser models with exchange terms, Calogero–{M}oser–{S}utherland Models ({M}ontr\'eal, {QC}, 1997), CRM Ser. Math. Phys., 37-51, (2000), Springer, New York [7] Baratta, Wendy, Some properties of {M}acdonald polynomials with prescribed symmetry, Kyushu Journal of Mathematics, 64, 2, 323-343, (2010) · Zbl 1200.33024 [8] Belbachir, H. and Boussicault, A. and Luque, J.-G., Hankel hyperdeterminants, rectangular {J}ack polynomials and even powers of the {V}andermonde, Journal of Algebra, 320, 11, 3911-3925, (2008) · Zbl 1168.33002 [9] Di Francesco, P. and Gaudin, M. and Itzykson, C. and Lesage, F., Laughlin’s wave functions, {C}oulomb gases and expansions of the discriminant, International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology, 9, 24, 4257-4351, (1994) · Zbl 0988.81563 [10] Forrester, Peter J. and Warnaar, S. Ole, The importance of the {S}elberg integral, American Mathematical Society. Bulletin. New Series, 45, 4, 489-534, (2008) · Zbl 1154.33002 [11] Gessel, Ira M. and Lv, Lun and Xin, Guoce and Zhou, Yue, A unified elementary approach to the {D}yson, {M}orris, {A}omoto, and {F}orrester constant term identities, Journal of Combinatorial Theory. Series A, 115, 8, 1417-1435, (2008) · Zbl 1207.05016 [12] Gustafson, Robert A., A generalization of {S}elberg’s beta integral, American Mathematical Society. Bulletin. New Series, 22, 1, 97-105, (1990) · Zbl 0693.33001 [13] Habsieger, Laurent, Une {\(q\)}-int\'egrale de {S}elberg et {A}skey, SIAM Journal on Mathematical Analysis, 19, 6, 1475-1489, (1988) · Zbl 0664.33001 [14] Hamada, Sayaka, Proof of {B}aker–{F}orrester’s constant term conjecture for the cases {\(N_1=2,3\)}, Kyushu Journal of Mathematics, 56, 2, 243-266, (2002) · Zbl 1010.33011 [15] Humphreys, James E., Reflection groups and {C}oxeter groups, Cambridge Studies in Advanced Mathematics, 29, xii+204, (1990), Cambridge University Press, Cambridge · Zbl 0725.20028 [16] Ito, Masahiko and Forrester, Peter J., A bilateral extension of the {\(q}-{S\)}elberg integral, Transactions of the American Mathematical Society, 369, 4, 2843-2878, (2017) · Zbl 1360.33016 [17] Kadell, Kevin W. J., A proof of {A}skey’s conjectured {\(q\)}-analogue of {S}elberg’s integral and a conjecture of {M}orris, SIAM Journal on Mathematical Analysis, 19, 4, 969-986, (1988) · Zbl 0643.33004 [18] Kaneko, Jyoichi, {\(q}-{S\)}elberg integrals and {M}acdonald polynomials, Annales Scientifiques de l’\'Ecole Normale Sup\'erieure. Quatri\`eme S\'erie, 29, 5, 583-637, (1996) · Zbl 0910.33011 [19] Kaneko, Jyoichi, On {B}aker–{F}orrester’s constant term conjecture, Journal of the Ramanujan Mathematical Society, 18, 4, 349-367, (2003) · Zbl 1229.33029 [20] K\'arolyi, Gyula and Nagy, Zolt\'an L\'or\'ant and Petrov, Fedor V. and Volkov, Vladislav, A new approach to constant term identities and {S}elberg-type integrals, Advances in Mathematics, 277, 252-282, (2015) · Zbl 1315.82013 [21] Koornwinder, Tom H., Askey–{W}ilson polynomials for root systems of type {\(BC\)}, Hypergeometric Functions on Domains of Positivity, {J}ack Polynomials, and Applications ({T}ampa, {FL}, 1991), Contemp. Math., 138, 189-204, (1992), Amer. Math. Soc., Providence, RI · Zbl 0797.33014 [22] Lusztig, George, Affine {H}ecke algebras and their graded version, Journal of the American Mathematical Society, 2, 3, 599-635, (1989) · Zbl 0715.22020 [23] Macdonald, I. G., The {P}oincar\'e series of a {C}oxeter group, Mathematische Annalen, 199, 161-174, (1972) · Zbl 0286.20062 [24] Macdonald, I. G., Affine {H}ecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, x+175, (2003), Cambridge University Press, Cambridge · Zbl 1024.33001 [25] Mehta, Madan Lal and Dyson, Freeman J., Statistical theory of the energy levels of complex systems. {V}, Journal of Mathematical Physics, 4, 713-719, (1963) · Zbl 0133.45202 [26] Morris, II, Walter Garfield, Constant term identities for finite and affine root systems, conjectures and theorems [27] Noumi, Masatoshi, Macdonald–{K}oornwinder polynomials and affine {H}ecke rings, S\={u}rikaisekikenky\={u}sho K\={o}ky\={u}roku, 919, 44-55, (1995) · Zbl 0900.05017 [28] Sahi, Siddhartha, Some properties of {K}oornwinder polynomials, {\(q\)}-Series from a Contemporary Perspective ({S}outh {H}adley, {MA}, 1998), Contemp. Math., 254, 395-411, (2000), Amer. Math. Soc., Providence, RI · Zbl 0959.33008 [29] Selberg, Atle, Remarks on a multiple integral, Norsk Matematisk Tidsskrift, 26, 71-78, (1944) · Zbl 0063.06870 [30] Stokman, Jasper V., On {\(BC\)} type basic hypergeometric orthogonal polynomials, Transactions of the American Mathematical Society, 352, 4, 1527-1579, (2000) · Zbl 0936.33008 [31] Stokman, Jasper V., Koornwinder polynomials and affine {H}ecke algebras, International Mathematics Research Notices, 2000, 19, 1005-1042, (2000) · Zbl 0965.33010 [32] Stokman, Jasper V., Lecture notes on {K}oornwinder polynomials, Laredo {L}ectures on {O}rthogonal {P}olynomials and {S}pecial {F}unctions, Adv. Theory Spec. Funct. Orthogonal Polynomials, 145-207, (2004), Nova Sci. Publ., Hauppauge, NY · Zbl 1083.33014 [33] Stokman, Jasper V. and Koornwinder, Tom H., Limit transitions for {BC} type multivariable orthogonal polynomials, Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques, 49, 2, 373-404, (1997) · Zbl 0881.33026 [34] Xin, Guoce and Zhou, Yue, A {L}aurent series proof of the {H}absieger–{K}adell {\(q\)}-{M}orris identity, Electronic Journal of Combinatorics, 21, 3, 3.38, 16 pages, (2014) · Zbl 1301.05037 [35] Warnaar, S. Ole, {\(q}-{S\)}elberg integrals and {M}acdonald polynomials, Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 10, 2, 237-268, (2005) · Zbl 1086.33018 [36] Warnaar, S. Ole, The {\({\mathfrak{sl}}_3} {S\)}elberg integral, Advances in Mathematics, 224, 2, 499-524, (2010) · Zbl 1194.33014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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