The importance of the Selberg integral. (English) Zbl 1154.33002
Following the recent death of Atle Selberg in 2007, this article concentrates on one of his many contributions to mathematics and reviews its ramifications in modern mathematical physics. The application in question is the so-called Selberg integral defined by
\[
\begin{aligned} S_n(\alpha,\beta,\gamma)&=\int_0^1 \cdots \int_0^1 \prod_{i=1}^nt_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1\leq i<j\leq n}|t_i-t_j|^{2\gamma}dt_1 \cdots dt_n\\ &=\prod_{j=0}^{n-1}\frac{\Gamma(\alpha+j\gamma) \Gamma(\beta+j\gamma)\Gamma(1+(j+1)\gamma)}{\Gamma( \alpha+\beta+(n+j-1)\gamma)\Gamma(1+\gamma)}\end{aligned}
\]
valid for complex parameters \(\alpha\), \(\beta\), \(\gamma\) such that the integral converges absolutely. The function \(S_n\) is an \(n\)-dimensional generalisation of the Euler beta integral. A. Selberg established this result in [Norsk Mat. Tidsskr. 26, 71–79 (1944; Zbl 0063.06870)] (in Norwegian) in 1944: he first dealt with the case of positive integer \(\gamma\) and then extended this result to complex values of the parameters by appeal to analytic continuation arguments.
The result then lay dormant until the 1970s when E. Bombieri contacted Selberg concerning a similar integral arising in prime number theory. The realisation that Selberg had established the functional form of the above integral immediately led to the resolution of Bombieri’s question and also of the Dyson-Mehta conjecture arising in random matrix theory made a decade or so earlier. The Morris integral and cases of the Macdonald conjectures made in the 1980s were also shown to be related to \(S_n\). This review article traces the development of integrals related to \(S_n\) and discusses the extension to \(q\)-analogues and to multivariable orthogonal polynomials (Jack polynomial theory).
The continued importance of the Selberg integral constitutes the final part of the paper. The authors discuss recent applications arising in mathematical physics in random matrix theory, Calogero-Sutherland quantum many-body systems, Knishnik-Zamolodchikov equations, elliptic Selberg integrals and, finally, in the value distributions of the Riemann \(\zeta(s) \) function on the critical line \(s=1/2+it\), \(t>0\).
The result then lay dormant until the 1970s when E. Bombieri contacted Selberg concerning a similar integral arising in prime number theory. The realisation that Selberg had established the functional form of the above integral immediately led to the resolution of Bombieri’s question and also of the Dyson-Mehta conjecture arising in random matrix theory made a decade or so earlier. The Morris integral and cases of the Macdonald conjectures made in the 1980s were also shown to be related to \(S_n\). This review article traces the development of integrals related to \(S_n\) and discusses the extension to \(q\)-analogues and to multivariable orthogonal polynomials (Jack polynomial theory).
The continued importance of the Selberg integral constitutes the final part of the paper. The authors discuss recent applications arising in mathematical physics in random matrix theory, Calogero-Sutherland quantum many-body systems, Knishnik-Zamolodchikov equations, elliptic Selberg integrals and, finally, in the value distributions of the Riemann \(\zeta(s) \) function on the critical line \(s=1/2+it\), \(t>0\).
Reviewer: R. B. Paris (Dundee)
MSC:
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 34M30 | Asymptotics and summation methods for ordinary differential equations in the complex domain |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 65D20 | Computation of special functions and constants, construction of tables |
Citations:
Zbl 0063.06870References:
| [1] | AMS-IMS-SIAM meeting Interactions of Random Matrix Theory, Integrable Systems, and Stochastic Processes, June 2007. |
| [2] | Greg W. Anderson, The evaluation of Selberg sums, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 8, 469 – 472 (English, with French summary). · Zbl 0724.11063 |
| [3] | Greg W. Anderson, A short proof of Selberg’s generalized beta formula, Forum Math. 3 (1991), no. 4, 415 – 417. · Zbl 0723.33002 · doi:10.1515/form.1991.3.415 |
| [4] | George E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 191 – 224. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. · Zbl 0342.33001 |
| [5] | George E. Andrews and Richard Askey, Another \?-extension of the beta function, Proc. Amer. Math. Soc. 81 (1981), no. 1, 97 – 100. · Zbl 0471.33001 |
| [6] | George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. · Zbl 0920.33001 |
| [7] | K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987), no. 2, 545 – 549. · Zbl 0639.33001 · doi:10.1137/0518042 |
| [8] | K. Aomoto, On the complex Selberg integral, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 385 – 399. · Zbl 0639.33002 · doi:10.1093/qmath/38.4.385 |
| [9] | Kazuhiko Aomoto, On a theta product formula for the symmetric \?-type connection function, Osaka J. Math. 32 (1995), no. 1, 35 – 39. · Zbl 0822.33011 |
| [10] | Kazuhiko Aomoto, On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), no. 2, 115 – 126. · Zbl 0918.33013 · doi:10.1023/A:1008629309210 |
| [11] | Richard Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), no. 6, 938 – 951. · Zbl 0458.33002 · doi:10.1137/0511084 |
| [12] | R. Askey, Letter to the SIAM minisymposium “Problems and solutions in special functions”, in: OP-SF NET 5.5 (Web resource), 1998. |
| [13] | Richard Askey and Donald Richards, Selberg’s second beta integral and an integral of Mehta, Probability, statistics, and mathematics, Academic Press, Boston, MA, 1989, pp. 27 – 39. · Zbl 0683.33001 |
| [14] | Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. · Zbl 0572.33012 · doi:10.1090/memo/0319 |
| [15] | T. H. Baker and P. J. Forrester, Finite-\? fluctuation formulas for random matrices, J. Statist. Phys. 88 (1997), no. 5-6, 1371 – 1386. · Zbl 0939.82020 · doi:10.1007/BF02732439 |
| [16] | T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys. 188 (1997), no. 1, 175 – 216. · Zbl 0903.33010 · doi:10.1007/s002200050161 |
| [17] | T. H. Baker and P. J. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), no. 1, 1 – 50. · Zbl 0948.33012 · doi:10.1215/S0012-7094-98-09501-1 |
| [18] | E.W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Phil. Soc. 19 (1904), 374-425. |
| [19] | Daniel Barsky and Michel Carpentier, Polynômes de Jacobi généralisés et intégrales de Selberg, Electron. J. Combin. 3 (1996), no. 2, Research Paper 1, approx. 33 pp., issn=1077-8926, review=\MR{1392486}, (French, with French summary). The Foata Festschrift. · Zbl 0883.33010 |
| [20] | R. J. Beerends and E. M. Opdam, Certain hypergeometric series related to the root system \?\?, Trans. Amer. Math. Soc. 339 (1993), no. 2, 581 – 609. · Zbl 0794.33009 |
| [21] | M. C. Bergère, Proof of Serban’s conjecture, J. Math. Phys. 39 (1998), no. 1, 30 – 46. · Zbl 0913.33006 · doi:10.1063/1.532331 |
| [22] | George Pólya, Collected papers, The MIT Press, Cambridge, Mass.-London, 1974. Vol. 1: Singularities of analytic functions; Edited by R. P. Boas; Mathematicians of Our Time, Vol. 7. |
| [23] | R.P. Boas Jr., Book Reviews: New Journals Bull. Amer. Math. Soc. 60 (1954), 92-93. |
| [24] | E. Bombieri, Private correspondence, 28 August 2007. |
| [25] | Alexei Borodin and Grigori Olshanski, \?-measures on partitions, Robinson-Schensted-Knuth correspondence, and \?=2 random matrix ensembles, Random matrix models and their applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 71 – 94. · Zbl 0987.15013 |
| [26] | F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196. |
| [27] | F. Carlson, Sur une Classe de Séries de Taylor, Ph.D. thesis, Uppsala Univ. 1914. · JFM 45.0645.01 |
| [28] | Ivan Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), no. 2, 411 – 431. · Zbl 0725.20012 · doi:10.1007/BF01243918 |
| [29] | Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191 – 216. · Zbl 0822.33008 · doi:10.2307/2118632 |
| [30] | Ivan Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol. 319, Cambridge University Press, Cambridge, 2005. · Zbl 1087.20003 |
| [31] | Ivan Cherednik and Viktor Ostrik, From double Hecke algebra to Fourier transform, Selecta Math. (N.S.) 9 (2003), no. 2, 161 – 249. · Zbl 1027.22014 |
| [32] | Paula Beazley Cohen and Amitai Regev, Asymptotics of multinomial sums and identities between multi-integrals, Israel J. Math. 112 (1999), 301 – 325. · Zbl 1026.05103 · doi:10.1007/BF02773486 |
| [33] | A. G. Constantine, Some non-central distribution problems in multivariate analysis, Ann. Math. Statist. 34 (1963), 1270 – 1285. · Zbl 0123.36801 · doi:10.1214/aoms/1177703863 |
| [34] | Hasan Coskun and Robert A. Gustafson, Well-poised Macdonald functions \?_{\?} and Jackson coefficients \?_{\?} on \?\?_{\?}, Jack, Hall-Littlewood and Macdonald polynomials, Contemp. Math., vol. 417, Amer. Math. Soc., Providence, RI, 2006, pp. 127 – 155. · Zbl 1154.05059 · doi:10.1090/conm/417/07919 |
| [35] | A. W. Davis, On the marginal distributions of the latent roots of the multivariate beta matrix, Ann. Math. Statist. 43 (1972), 1664 – 1670. · Zbl 0252.62028 · doi:10.1214/aoms/1177692399 |
| [36] | Amédée Debiard, Système différentiel hypergéométrique et parties radiales des opérateurs invariants des espaces symétriques de type \?\?_{\?}, Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986) Lecture Notes in Math., vol. 1296, Springer, Berlin, 1987, pp. 42 – 124. · doi:10.1007/BFb0078523 |
| [37] | J. F. van Diejen, Self-dual Koornwinder-Macdonald polynomials, Invent. Math. 126 (1996), no. 2, 319 – 339. · Zbl 0869.33012 · doi:10.1007/s002220050102 |
| [38] | J. F. van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Comm. Math. Phys. 188 (1997), no. 2, 467 – 497. · Zbl 0917.33008 · doi:10.1007/s002200050174 |
| [39] | J. F. van Diejen and V. P. Spiridonov, Elliptic Selberg integrals, Internat. Math. Res. Notices 20 (2001), 1083 – 1110. · Zbl 1010.33010 · doi:10.1155/S1073792801000526 |
| [40] | A.L. Dixon, Generalizations of Legendre’s formula \( KE'-(K-E)K'=\frac{1}{2}\pi\), Proc. London Math. Soc. 3 (1905), 206-224. · JFM 36.0506.01 |
| [41] | Vl. S. Dotsenko and V. A. Fateev, Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge \?\le 1, Nuclear Phys. B 251 (1985), no. 5-6, 691 – 734. , https://doi.org/10.1016/S0550-3213(85)80004-3 Vl. S. Dotsenko and V. A. Fateev, Operator algebra of two-dimensional conformal theories with central charge \?\le 1, Phys. Lett. B 154 (1985), no. 4, 291 – 295. · doi:10.1016/0370-2693(85)90366-1 |
| [42] | Ioana Dumitriu and Alan Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830 – 5847. · Zbl 1060.82020 · doi:10.1063/1.1507823 |
| [43] | Charles F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167 – 183. · Zbl 0652.33004 |
| [44] | Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. · Zbl 0964.33001 |
| [45] | Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140 – 156. · Zbl 0105.41604 · doi:10.1063/1.1703773 |
| [46] | Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs, vol. 58, American Mathematical Society, Providence, RI, 1998. · Zbl 0903.17006 |
| [47] | L. Euler, De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt, Comm. Acad. Sci. Petropolitanae 5 (1730), 36-57. |
| [48] | L. Euler, Institutiones Calculi Integralis, II, Opera Omnia, Ser. 1, Vol. 12. · JFM 44.0011.01 |
| [49] | Ronald J. Evans, Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), no. 3-4, 197 – 209 (1982). · Zbl 0491.12020 |
| [50] | Ronald J. Evans, The evaluation of Selberg character sums, Enseign. Math. (2) 37 (1991), no. 3-4, 235 – 248. · Zbl 0774.11046 |
| [51] | Ronald J. Evans, Multidimensional \?-beta integrals, SIAM J. Math. Anal. 23 (1992), no. 3, 758 – 765. · Zbl 0766.33013 · doi:10.1137/0523039 |
| [52] | 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. · Zbl 0820.33001 · doi:10.1090/conm/166/01631 |
| [53] | Ronald J. Evans, Selberg-Jack character sums of dimension 2, J. Number Theory 54 (1995), no. 1, 1 – 11. · Zbl 0883.11051 · doi:10.1006/jnth.1995.1097 |
| [54] | R.J. Evans, Private correspondence, 31 August 2007. |
| [55] | Ronald J. Evans and William A. Root, Conjectures for Selberg character sums, J. Ramanujan Math. Soc. 3 (1988), no. 1, 111 – 128. · Zbl 0683.12011 |
| [56] | G. Felder, L. Stevens, and A. Varchenko, Elliptic Selberg integrals and conformal blocks, Math. Res. Lett. 10 (2003), no. 5-6, 671 – 684. · Zbl 1041.33003 · doi:10.4310/MRL.2003.v10.n5.a10 |
| [57] | P. J. Forrester, Recurrence equations for the computation of correlations in the 1/\?² quantum many-body system, J. Statist. Phys. 72 (1993), no. 1-2, 39 – 50. · Zbl 1096.82504 · doi:10.1007/BF01048039 |
| [58] | P.J. Forrester, Log-Gases and Random Matrices, http://www.ms.unimelb.edu.au/ matpjf · Zbl 1217.82003 |
| [59] | P.J. Forrester, Beta random matrix ensembles, to appear, Proceedings of the IMS (Singapore) programme on Random Matrix Theory and its Applications to Statistics and Wireless Communications. · Zbl 1182.15020 |
| [60] | P.J. Forrester, A random matrix decimation procedure relating \( \beta = 2/(r+1)\) to \( \beta = 2(r+1)\), to appear in Commun. Math. Phys.; arXiv:0711.1914. · Zbl 1173.15011 |
| [61] | Peter J. Forrester and Eric M. Rains, Interpretations of some parameter dependent generalizations of classical matrix ensembles, Probab. Theory Related Fields 131 (2005), no. 1, 1 – 61. · Zbl 1056.05142 · doi:10.1007/s00440-004-0375-6 |
| [62] | P.J. Forrester and E.M. Rains, in preparation. |
| [63] | Frank G. Garvan, Some Macdonald-Mehta integrals by brute force, \?-series and partitions (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 18, Springer, New York, 1989, pp. 77 – 98. · Zbl 0697.33008 · doi:10.1007/978-1-4684-0637-5_8 |
| [64] | F.G. Garvan, Unpublished computer proof of the \( k=1\) case of (1.20) for H\( _4\). |
| [65] | F. G. Garvan, A proof of the Macdonald-Morris root system conjecture for \?\(_{4}\), SIAM J. Math. Anal. 21 (1990), no. 3, 803 – 821. · Zbl 0711.33001 · doi:10.1137/0521045 |
| [66] | Frank G. Garvan and Gaston Gonnet, Macdonald’s constant term conjectures for exceptional root systems, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 343 – 347. · Zbl 0737.33011 |
| [67] | George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. · Zbl 1129.33005 |
| [68] | A.O. Gelfond, Sur un theorème de M.G. Polya, Atti Reale Accad. Naz Lincei 10 (1929), 569-574. · JFM 55.0778.01 |
| [69] | I. J. Good, Short proof of a conjecture by Dyson, J. Mathematical Phys. 11 (1970), 1884. · doi:10.1063/1.1665339 |
| [70] | J. Gunson, Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Mathematical Phys. 3 (1962), 752 – 753. · Zbl 0111.43903 · doi:10.1063/1.1724277 |
| [71] | Robert A. Gustafson, A generalization of Selberg’s beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 97 – 105. · Zbl 0693.33001 |
| [72] | Robert A. Gustafson, Some \?-beta integrals on \?\?(\?) and \?\?(\?) that generalize the Askey-Wilson and Nasrallah-Rahman integrals, SIAM J. Math. Anal. 25 (1994), no. 2, 441 – 449. · Zbl 0819.33009 · doi:10.1137/S0036141092248614 |
| [73] | Laurent Habsieger, La \?-conjecture de Macdonald-Morris pour \?\(_{2}\), C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 6, 211 – 213 (French, with English summary). · Zbl 0598.05006 |
| [74] | Laurent Habsieger, Une \?-intégrale de Selberg et Askey, SIAM J. Math. Anal. 19 (1988), no. 6, 1475 – 1489 (French, with English summary). · Zbl 0664.33001 · doi:10.1137/0519111 |
| [75] | G. H. Hardy and J. E. Littlewood, Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1916), no. 1, 119 – 196. · JFM 46.0498.01 · doi:10.1007/BF02422942 |
| [76] | G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), no. 2, 341 – 350. · Zbl 0721.33009 · doi:10.1007/BF01239517 |
| [77] | Carl S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474 – 523. · Zbl 0066.32002 · doi:10.2307/1969810 |
| [78] | L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translations of Mathematical Monographs, vol. 6, American Mathematical Society, Providence, R.I., 1979. Translated from the Russian, which was a translation of the Chinese original, by Leo Ebner and Adam Korányi; With a foreword by M. I. Graev; Reprint of the 1963 edition. · Zbl 0507.32025 |
| [79] | Institute for Advanced Study, press releases: Atle Selberg 1917-2007, http:/www.ias.edu/newsroom/announcements/view/1186683853.html |
| [80] | Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. · Zbl 1082.42016 |
| [81] | 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. · Zbl 0902.33011 · doi:10.1006/jmaa.1997.5665 |
| [82] | Kurt Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437 – 476. · Zbl 0969.15008 · doi:10.1007/s002200050027 |
| [83] | Kurt Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002), no. 2, 225 – 280. · Zbl 1008.60019 · doi:10.1007/s004400100187 |
| [84] | Kevin W. J. Kadell, A proof of some \?-analogues of Selberg’s integral for \?=1, SIAM J. Math. Anal. 19 (1988), no. 4, 944 – 968. , https://doi.org/10.1137/0519066 Kevin W. J. Kadell, A proof of Askey’s conjectured \?-analogue of Selberg’s integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), no. 4, 969 – 986. · Zbl 0643.33004 · doi:10.1137/0519067 |
| [85] | Kevin W. J. Kadell, A proof of some \?-analogues of Selberg’s integral for \?=1, SIAM J. Math. Anal. 19 (1988), no. 4, 944 – 968. , https://doi.org/10.1137/0519066 Kevin W. J. Kadell, A proof of Askey’s conjectured \?-analogue of Selberg’s integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), no. 4, 969 – 986. · Zbl 0643.33004 · doi:10.1137/0519067 |
| [86] | Kevin W. J. Kadell, An integral for the product of two Selberg-Jack symmetric polynomials, Compositio Math. 87 (1993), no. 1, 5 – 43. · Zbl 0788.33007 |
| [87] | Kevin W. J. Kadell, A proof of the \?-Macdonald-Morris conjecture for \?\?_{\?}, Mem. Amer. Math. Soc. 108 (1994), no. 516, vi+80. · Zbl 0796.17003 · doi:10.1090/memo/0516 |
| [88] | Kevin W. J. Kadell, The Selberg-Jack symmetric functions, Adv. Math. 130 (1997), no. 1, 33 – 102. · Zbl 0885.33009 · doi:10.1006/aima.1997.1642 |
| [89] | Saburo Kakei, Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials, J. Math. Phys. 39 (1998), no. 9, 4993 – 5006. · Zbl 1050.33008 · doi:10.1063/1.532505 |
| [90] | Jyoichi Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), no. 4, 1086 – 1110. · Zbl 0783.33008 · doi:10.1137/0524064 |
| [91] | Jyoichi Kaneko, \?-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 5, 583 – 637. · Zbl 0910.33011 |
| [92] | S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc. No. 12 (1953), 93. · Zbl 0052.18501 |
| [93] | Nicholas M. Katz and Peter Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 1 – 26. · Zbl 0921.11047 |
| [94] | J. P. Keating and N. C. Snaith, Random matrix theory and \?(1/2+\?\?), Comm. Math. Phys. 214 (2000), no. 1, 57 – 89. · Zbl 1051.11048 · doi:10.1007/s002200000261 |
| [95] | J. P. Keating and N. C. Snaith, Random matrix theory and \?-functions at \?=1/2, Comm. Math. Phys. 214 (2000), no. 1, 91 – 110. · Zbl 1051.11047 · doi:10.1007/s002200000262 |
| [96] | J. P. Keating and N. C. Snaith, Random matrices and \?-functions, J. Phys. A 36 (2003), no. 12, 2859 – 2881. Random matrix theory. · Zbl 1074.11053 · doi:10.1088/0305-4470/36/12/301 |
| [97] | J. P. Keating, N. Linden, and Z. Rudnick, Random matrix theory, the exceptional Lie groups and \?-functions, J. Phys. A 36 (2003), no. 12, 2933 – 2944. Random matrix theory. · Zbl 1074.11052 · doi:10.1088/0305-4470/36/12/305 |
| [98] | Rowan Killip and Irina Nenciu, Matrix models for circular ensembles, Int. Math. Res. Not. 50 (2004), 2665 – 2701. · Zbl 1255.82004 · doi:10.1155/S1073792804141597 |
| [99] | V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), no. 1, 83 – 103. · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2 |
| [100] | Tom H. Koornwinder, Askey-Wilson polynomials for root systems of type \?\?, 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. · Zbl 0797.33014 · doi:10.1090/conm/138/1199128 |
| [101] | Adam Korányi, Hua-type integrals, hypergeometric functions and symmetric polynomials, International Symposium in Memory of Hua Loo Keng, Vol. II (Beijing, 1988) Springer, Berlin, 1991, pp. 169 – 180. · Zbl 0814.33009 |
| [102] | Michel Lassalle, Polynômes de Jacobi généralisés, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 6, 425 – 428 (French, with English summary). · Zbl 0742.33005 |
| [103] | Michel Lassalle, Polynômes de Laguerre généralisés, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 10, 725 – 728 (French, with English summary). · Zbl 0739.33007 |
| [104] | Michel Lassalle, Polynômes de Hermite généralisés, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 9, 579 – 582 (French, with English summary). · Zbl 0748.33006 |
| [105] | Jean-Gabriel Luque and Jean-Yves Thibon, Hankel hyperdeterminants and Selberg integrals, J. Phys. A 36 (2003), no. 19, 5267 – 5292. · Zbl 1058.33019 · doi:10.1088/0305-4470/36/19/306 |
| [106] | J.-G. Luque and J.-Y. Thibon, Hyperdeterminantal calculations of Selberg’s and Aomoto’s integrals, Molecular Physics 102 (2004), 1351-1359. |
| [107] | B. Nassrallah and Mizan Rahman, Projection formulas, a reproducing kernel and a generating function for \?-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), no. 1, 186 – 197. · Zbl 0564.33009 · doi:10.1137/0516014 |
| [108] | I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988 – 1007. , https://doi.org/10.1137/0513070 Richard Askey, A \?-beta integral associated with \?\?\(_{1}\), SIAM J. Math. Anal. 13 (1982), no. 6, 1008 – 1010. · Zbl 0501.33002 · doi:10.1137/0513071 |
| [109] | I. G. Macdonald, Commuting differential operators and zonal spherical functions, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 189 – 200. · doi:10.1007/BFb0079238 |
| [110] | I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. · Zbl 0899.05068 |
| [111] | I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. · Zbl 1024.33001 |
| [112] | I. G. Macdonald, Hypergeometric functions, unpublished manuscript. · Zbl 0035.20801 |
| [113] | M. L. Mehta, Random matrices and the statistical theory of energy levels, Academic Press, New York-London, 1967. · Zbl 0925.60011 |
| [114] | M.L. Mehta, Problem 74-6, Three multiple integrals, SIAM Review 16 (1974), 256-257. |
| [115] | Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. · Zbl 1107.15019 |
| [116] | Madan Lal Mehta and Freeman J. Dyson, Statistical theory of the energy levels of complex systems. V, J. Mathematical Phys. 4 (1963), 713 – 719. · Zbl 0133.45202 · doi:10.1063/1.1704009 |
| [117] | Katsuhisa Mimachi, Reducibility and irreducibility of the Gauss-Manin system associated with a Selberg type integral, Nagoya Math. J. 132 (1993), 43 – 62. · Zbl 0833.33012 |
| [118] | K. Mimachi, The connection problem associated with a Selberg type integral and the \( q\)-Racah polynomials, arXiv:0710.2167. · Zbl 1285.34077 |
| [119] | Katsuhisa Mimachi and Masaaki Yoshida, The reciprocity relation of the Selberg function, Proceedings of the International Conference on Special Functions and their Applications (Chennai, 2002), 2003, pp. 209 – 215. · Zbl 1069.33004 · doi:10.1016/S0377-0427(03)00623-X |
| [120] | Katsuhisa Mimachi and Masaaki Yoshida, Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory, Comm. Math. Phys. 250 (2004), no. 1, 23 – 45. · Zbl 1069.32015 · doi:10.1007/s00220-004-1138-z |
| [121] | Katsuhisa Mimachi and Takashi Takamuki, A generalization of the beta integral arising from the Knizhnik-Zamolodchikov equation for the vector representations of types \?_{\?}, \?_{\?} and \?_{\?}, Kyushu J. Math. 59 (2005), no. 1, 117 – 126. · Zbl 1088.33008 · doi:10.2206/kyushujm.59.117 |
| [122] | W.G. Morris, Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems, Ph.D. thesis, Univ. Wisconsin-Madison, 1982. |
| [123] | R. J. Muirhead, Systems of partial differential equations for hypergeometric functions of matrix argument, Ann. Math. Statist. 41 (1970), 991 – 1001. · Zbl 0225.62078 · doi:10.1214/aoms/1177696975 |
| [124] | Evgeny Mukhin and Alexander Varchenko, Remarks on critical points of phase functions and norms of Bethe vectors, Arrangements — Tokyo 1998, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000, pp. 239 – 246. · Zbl 1040.17001 |
| [125] | A.M. Odlyzko, The \( 10^{20}\)th zero of the Riemann zeta function and \( 70\) million of its neighbours, http://www.dtc.umn.edu/ odlyzko/unpublished/index.html |
| [126] | Andrei Okounkov, (Shifted) Macdonald polynomials: \?-integral representation and combinatorial formula, Compositio Math. 112 (1998), no. 2, 147 – 182. · Zbl 0897.05085 · doi:10.1023/A:1000436921311 |
| [127] | A. Okounkov, \?\?-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), no. 2, 181 – 207. · Zbl 0941.17005 · doi:10.1007/BF01236432 |
| [128] | Andrei Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), no. 1, 57 – 81. · Zbl 0986.05102 · doi:10.1007/PL00001398 |
| [129] | E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1 – 18. · Zbl 0696.33006 · doi:10.1007/BF01388841 |
| [130] | E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), no. 3, 333 – 373. · Zbl 0778.33009 |
| [131] | Eric M. Opdam, Lecture notes on Dunkl operators for real and complex reflection groups, MSJ Memoirs, vol. 8, Mathematical Society of Japan, Tokyo, 2000. With a preface by Toshio Oshima. · Zbl 0984.33001 |
| [132] | George Pólya, Collected papers, The MIT Press, Cambridge, Mass.-London, 1974. Vol. 1: Singularities of analytic functions; Edited by R. P. Boas; Mathematicians of Our Time, Vol. 7. |
| [133] | Mizan Rahman, An integral representation of a \(_{1}\)\(_{0}\)\?\(_{9}\) and continuous bi-orthogonal \(_{1}\)\(_{0}\)\?\(_{9}\) rational functions, Canad. J. Math. 38 (1986), no. 3, 605 – 618. · Zbl 0599.33015 · doi:10.4153/CJM-1986-030-6 |
| [134] | E.M. Rains, Transformations of elliptic hypergeometric integrals, to appear in Ann. of Math.; arXiv:math.QA/0309252. · Zbl 1209.33014 |
| [135] | E.M. Rains, Private communication. |
| [136] | E.M. Rains, Limits of elliptic hypergeometric integrals, to appear in Ramanujan J.; arXiv:math.CA/0607093. · Zbl 1178.33021 |
| [137] | E.M. Rains and V.P. Spiridonov, Determinants of elliptic hypergeometric integrals, to appear in Funct. Anal. Appl.; arXiv:0712.4253 · Zbl 1193.33201 |
| [138] | Amitai Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), no. 2, 115 – 136. · Zbl 0509.20009 · doi:10.1016/0001-8708(81)90012-8 |
| [139] | A. Regev, Private correspondence, 30 August 2007. |
| [140] | Donald Richards and Qifu Zheng, Determinants of period matrices and an application to Selberg’s multidimensional beta integral, Adv. in Appl. Math. 28 (2002), no. 3-4, 602 – 633. Special issue in memory of Rodica Simion. · Zbl 1011.33014 · doi:10.1006/aama.2001.0798 |
| [141] | Michael Rubinstein, Computational methods and experiments in analytic number theory, Recent perspectives in random matrix theory and number theory, London Math. Soc. Lecture Note Ser., vol. 322, Cambridge Univ. Press, Cambridge, 2005, pp. 425 – 506. · Zbl 1168.11329 · doi:10.1017/CBO9780511550492.015 |
| [142] | S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), no. 2, 1069 – 1146. · Zbl 0877.39002 · doi:10.1063/1.531809 |
| [143] | Vadim V. Schechtman and Alexander N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), no. 1, 139 – 194. · Zbl 0754.17024 · doi:10.1007/BF01243909 |
| [144] | Atle Selberg, Über einen Satz von A. Gelfond, Arch. Math. Naturvid. 44 (1941), 159 – 170 (German). · JFM 67.0270.01 |
| [145] | Atle Selberg, Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71 – 78 (Norwegian). · Zbl 0063.06870 |
| [146] | Atle Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), no. 5, 89 – 155. · Zbl 0061.08402 |
| [147] | Atle Selberg, Collected papers. Vol. I, Springer-Verlag, Berlin, 1989. With a foreword by K. Chandrasekharan. · Zbl 0675.10001 |
| [148] | V. P. Spiridonov, On the elliptic beta function, Uspekhi Mat. Nauk 56 (2001), no. 1(337), 181 – 182 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 1, 185 – 186. · Zbl 0997.33009 · doi:10.1070/rm2001v056n01ABEH000374 |
| [149] | V. P. Spiridonov, Elliptic beta integrals and special functions of hypergeometric type, Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 35, Kluwer Acad. Publ., Dordrecht, 2001, pp. 305 – 313. · Zbl 0997.33010 |
| [150] | V. P. Spiridonov, Short proofs of the elliptic beta integrals, Ramanujan J. 13 (2007), no. 1-3, 265 – 283. · Zbl 1118.33008 · doi:10.1007/s11139-006-0252-2 |
| [151] | V.P. Spiridonov, Elliptic hypergeometric functions, A complement to [6], written for its Russian edition; arXiv:0704.3099. · Zbl 1480.33009 |
| [152] | Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76 – 115. · Zbl 0743.05072 · doi:10.1016/0001-8708(89)90015-7 |
| [153] | Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. · Zbl 0889.05001 |
| [154] | R.P. Stanley, Queue problems revisited, Suomen Tehtäväniekat 59 (2005), 193-203. |
| [155] | R.P. Stanley, Private correspondence, 15 September 2007. |
| [156] | John R. Stembridge, A short proof of Macdonald’s conjecture for the root systems of type \?, Proc. Amer. Math. Soc. 102 (1988), no. 4, 777 – 786. · Zbl 0645.10017 |
| [157] | Jasper V. Stokman, Multivariable big and little \?-Jacobi polynomials, SIAM J. Math. Anal. 28 (1997), no. 2, 452 – 480. · Zbl 0892.33010 · doi:10.1137/S0036141095287192 |
| [158] | Jasper V. Stokman, On \?\? type basic hypergeometric orthogonal polynomials, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1527 – 1579. · Zbl 0936.33008 |
| [159] | B. Sutherland, Exact results for a quantum many body problem in one-dimension, Phys. Rev. A 4 (1971), 2019-2021. |
| [160] | B. Sutherland, Exact results for a quantum many-body problem in one dimension: II, Phys. Rev. A 5 (1972), 1372-1376. |
| [161] | V. Tarasov and A. Varchenko, Difference equations compatible with trigonometric KZ differential equations, Internat. Math. Res. Notices 15 (2000), 801 – 829. · Zbl 0971.39009 · doi:10.1155/S1073792800000441 |
| [162] | V. Tarasov and A. Varchenko, Selberg-type integrals associated with \?\?\(_{3}\), Lett. Math. Phys. 65 (2003), no. 3, 173 – 185. · Zbl 1055.33016 · doi:10.1023/B:MATH.0000010712.67685.9d |
| [163] | A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1206 – 1235, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 543 – 571. A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 146 – 158, 222 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 155 – 167. · Zbl 0695.33004 |
| [164] | A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1206 – 1235, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 543 – 571. A. N. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. II, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 146 – 158, 222 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 155 – 167. · Zbl 0695.33004 |
| [165] | Alexander Varchenko, Special functions, KZ type equations, and representation theory, CBMS Regional Conference Series in Mathematics, vol. 98, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2003. · Zbl 1053.33001 |
| [166] | S. Ole Warnaar, \?-Selberg integrals and Macdonald polynomials, Ramanujan J. 10 (2005), no. 2, 237 – 268. · Zbl 1086.33018 · doi:10.1007/s11139-005-4849-7 |
| [167] | S. Ole Warnaar, On the generalised Selberg integral of Richards and Zheng, Adv. in Appl. Math. 40 (2008), no. 2, 212 – 218. · Zbl 1134.33003 · doi:10.1016/j.aam.2006.12.002 |
| [168] | S.O. Warnaar, Bisymmetric functions, Macdonald polynomials and \( \mathfrak{sl}_3\) basic hypergeometric series, Compositio Math. 114 (2008), 271-303. · Zbl 1182.05129 |
| [169] | S.O. Warnaar, A Selberg integral for the Lie algebra A\( _n\), to appear in Acta Math.; arXiv:0708.1193. · Zbl 1243.33053 |
| [170] | S.O. Warnaar, The Mukhin-Varchenko conjecture for type A, to appear in the Proceedings of FPSAC 2008, Discrete Math. Theor. Comput. Sci. · Zbl 1394.33003 |
| [171] | Kenneth G. Wilson, Proof of a conjecture by Dyson, J. Mathematical Phys. 3 (1962), 1040 – 1043. · Zbl 0113.21403 · doi:10.1063/1.1724291 |
| [172] | E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. · JFM 53.0180.04 |
| [173] | Zhi Min Yan, A class of generalized hypergeometric functions in several variables, Canad. J. Math. 44 (1992), no. 6, 1317 – 1338. · Zbl 0769.33014 · doi:10.4153/CJM-1992-079-x |
| [174] | Doron Zeilberger and David M. Bressoud, A proof of Andrews’ \?-Dyson conjecture, Discrete Math. 54 (1985), no. 2, 201 – 224. · Zbl 0565.33001 · doi:10.1016/0012-365X(85)90081-0 |
| [175] | Doron Zeilberger, A proof of the \?\(_{2}\) case of Macdonald’s root system-Dyson conjecture, SIAM J. Math. Anal. 18 (1987), no. 3, 880 – 883. · Zbl 0643.05004 · doi:10.1137/0518065 |
| [176] | Doron Zeilberger, A unified approach to Macdonald’s root-system conjectures, SIAM J. Math. Anal. 19 (1988), no. 4, 987 – 1013. · Zbl 0658.05005 · doi:10.1137/0519068 |
| [177] | Doron Zeilberger, A Stembridge-Stanton style elementary proof of the Habsieger-Kadell \?-Morris identity, Discrete Math. 79 (1990), no. 3, 313 – 322. · Zbl 0693.05009 · doi:10.1016/0012-365X(90)90338-I |
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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
