Theta hypergeometric integrals

St. Petersburg Mathematical Journal

Volumn 15, Issue 6, 2004, Pages 929-967

  Spiridonov, V P ^a  

^a JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

Author keywords

Elliptic beta integral; Jacobi theta functions; Multiple hypergeometric integrals; Root system

Indexed keywords

EID: 85009787071     PISSN: 10610022     EISSN: 15477371     Source Type: Journal    
DOI: 10.1090/S1061-0022-04-00839-8     Document Type: Article

References (70)

1. G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia Math. Appl., vol. 71, Cambridge Univ. Press, Cambridge, 1999.MR1688958 (2000g:33001)
2. R. Askey, Beta integrals in Ramanujan’s papers, his unpublished work and further examples, Ramanujan Revisited (Urbana–Champaign, Ill., 1987), Acad. Press, Boston, MA, 1988, pp. 561–590.MR0938979 (89i:01044)
3. R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319.MR0783216 (87a:05023)
4. E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374–425.
5. E. W. Barnes, The linear difference equation of the first order, Proc. London Math. Soc. (2) 2 (1905), 438–469.
6. R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193–228.MR0290733 (44:7912)
7. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vols. I, II, III, McGraw-Hill Book Co., Inc., New York, etc., 1953, 1955.MR0058756 (15:419i);MR0066496 (16:586c)
8. E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable SOS models: local height probabilities and theta function identities, Nuclear Phys. B 290 (1987), 231–273.MR0910849 (89d:82114)
9. E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable SOS models. II. Proof of the star-triangle relation and combinatorial identities, Conformal Field Theory and Solvable Lattice Models (Kyoto, 1986), Adv. Stud. Pure Math., vol. 16, Acad. Press, Boston, MA, 1988, pp. 17–122.MR0972992 (90e:82077)
10. R. Y. Denis and R. A. Gustafson, An SU(n) q-beta integral transformation and multiple hypergeometric series identities, SIAMJ. Math. Anal. 23 (1992), 552–561.MR1147877 (92m:33031)
11. J. F. van Diejen and V. P. Spiridonov, An elliptic Macdonald–Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), 729–746.MR1809297 (2002m:33022)
12. J. F. van Diejen and V. P. Spiridonov, Elliptic Selberg integrals, Internat. Math. Res. Notices 2001, no. 20, 1083–1110.MR1857597 (2002j:33016)
13. J. F. van Diejen and V. P. Spiridonov, Modular hypergeometric residue sums of elliptic Selberg integrals, Lett. Math. Phys. 58 (2001), 223–238 (2002).MR1892922 (2003g:11046)
14. J. F. van Diejen and V. P. Spiridonov, Elliptic beta integrals and modular hypergeometric sums: an overview, Conference on Special Functions (Tempe, AZ, 2000), Rocky Mountain J. Math. 32 (2002), 639–656.MR1934909 (2003j:33061)
15. M. Eichler and D. Zagier, The theory of Jacobi forms, Progr. Math., vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985.MR0781735 (86j:11043)
16. L. D. Faddeev, Discrete Heisenberg–Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249–254
17. Modular double of a quantum group, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer, Dordrecht, 2000, pp. 149–156.MR1345554 (96i:46075);MR1805888 (2002a:81126)
18. 3, Adv. Math. 156 (2000), 44–76.MR1800253 (2002e:11058)
19. I. B. Frenkel and V. G. Turaev, Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions, The Arnold–Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 171–204.MR1429892 (98k:33034)
20. G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia Math. Appl., vol. 35, Cambridge Univ. Press, Cambridge, 1990.MR1052153 (91d:33034)
21. I. M. Gel′fand, M. I. Graev, and V. S. Petakh, General hypergeometric systems of equations and series of hypergeometric type, Uspekhi Mat. Nauk 47 (1992), no. 4, 3–82
22. English transl., Russian Math. Surveys 47 (1992), no. 4, 1–88.MR1208882 (94h:33005)
23. D. P. Gupta and D. R. Masson, Contiguous relations, continued fractions and orthogonality, Trans. Amer. Math. Soc. 350 (1998), 769–808.MR1407490 (99c:33028)
24. R. A. Gustafson, Some q-beta and Mellin–Barnes integrals with many parameters associated to the classical groups, SIAM J. Math. Anal. 23 (1992), 525–551.MR1147876 (92m:33006)
25. R. A. Gustafson, Some q-beta integrals on SU(n) and Sp(n) that generalize the Askey–Wilson and Nassrallah–Rahman integrals, SIAM J. Math. Anal. 25 (1994), 441–449.MR1266569 (95b:33053)
26. R. A. Gustafson and C. Krattenthaler, Determinant evaluations and U(n) extensions of Heine’s 2ϕ1-transformations, Special Functions, q-Series and Related Topics (Toronto, ON, 1955), Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 83–89.MR1448682 (98d:33009)
27. R. A. Gustafson and M. A. Rakha, q-beta integrals and multivariate basic hypergeometric series associated to root systems of type Am, Conference on Combinatorics and Physics (Los Alamos, NM, 1998), Ann. Comb. 4 (2000), 347–373.MR1811060 (2003a:33034)
28. W. J. Holman III, L. C. Biedenharn, and J. D. Louck, On hypergeometric series well-poised in SU(n), SIAM J. Math. Anal. 7 (1976), 529–541.MR0412504 54:627
29. M. E. H. Ismail and D. R. Masson, Generalized orthogonality and continued fractions, J. Approx. Theory 83 (1995), 1–40.MR1354960 (97d:42020)
30. M. E. H. Ismail and M. Rahman, The associated Askey–Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), 201–237.MR1013333 (92c:33019)
31. F. H. Jackson, The basic gamma-function and the elliptic functions, Proc. Roy. Soc. London Ser. A 76 (1905), 127–144.
32. M. Jimbo and T. Miwa, Quantum KZ equation with q = 1 and correlation functions of the XXZ model in the gapless regime, J. Phys. A: Math. Gen. 29 (1996), 2923–2958.MR1398600 (97i:82021)
33. S. Kharchev, D. Lebedev, and M. Semenov-Tian-Shansky, Unitary representations of Uq(sl(2, R)), the modular double and the multiparticle q-deformed Toda chains, Comm. Math. Phys. 225 (2002), 573–609.MR1888874 (2003b:17019)
34. T. 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.MR1199128 (94e:33039)
35. C. Krattenthaler, The major counting of nonintersecting lattice paths and generating functions for tableaux, Mem. Amer. Math. Soc. 115 (1995), no. 552.MR1254150 (95i:05109)
36. I. G. Macdonald, Constant term identities, orthogonal polynomials, and affine Hecke algebras, Proceedings of the International Congress of Mathematicians, Vol. 1 (Berlin, 1998), Doc. Math. 1998, Extra Vol., 303–317.MR1648036 (2000b:33007)
37. S. C. Milne, The multidimensional 1Ψ1 sum and Macdonald identities for A(1) l, Theta Functions — Bowdoin 1987, Part 2 (Brunswick, ME, 1987), Proc. Sympos. Pure Math., vol. 49, Part 2, Amer. Math. Soc., Providence, RI, 1989, pp. 323–359.MR013180 (91f:33011)
38. S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Ramanujan J. 6 (2002), 7–149.MR1906722 (2003m:11060)
39. S. C. Milne and G. M. Lilly, Consequences of the Al and Cl Bailey transform and Bailey lemma, Discrete Math. 139 (1995), 319–346.MR1336846 (96g:33025)
40. B. Nassrallah and M. Rahman, Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), 186–197.MR0772878 (87b:33009)
41. M. Nishizawa and K. Ueno, Integral solutions of hypergeometric q-difference systems with q = 1, Physics and Combinatorics (Nagoya, 1999), World Sci. Publishing, River Edge, NJ, 2001, pp. 273–286.MR1865041 (2002h:39026)
42. M. Rahman, An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions, Canad. J. Math. 38 (1986), 605–618.MR0845667 (87i:33011)
43. M. Rahman, Biorthogonality of a system of rational functions with respect to a positive measure on [−1, 1], SIAM J. Math. Anal. 22 (1991), 1430–1441.MR1112517 (92h:33016)
44. M. Rahman and S. K. Suslov, Classical biorthogonal rational functions, Methods of Approximation Theory in Complex Analysis and Mathematical Physics (Leningrad, 1991), Lecture Notes in Math., vol. 1550, Springer-Verlag, Berlin, 1993, pp. 131–146.MR1322295 (95m:42034)
45. M. Rahman and S. K. Suslov, The Pearson equation and the beta integrals, SIAMJ.Math. Anal. 25 (1994), 646–693.MR1266583 (95f:33001)
46. H. Rosengren, A proof of a multivariable elliptic summation formula conjectured by Warnaar, q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 193–202.MR1874532 (2002m:33027)
47. H. Rosengren, Elliptic hypergeometric series on root systems, Adv. Math. 81 (2004), 417–447.MR2026866
48. S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069–1146.MR1434226 (98m:58065)
49. S. N. M. Ruijsenaars, A generalized hypergeometric function satisfying four analytic difference equations of Askey–Wilson type, Comm. Math. Phys. 206 (1999), 639–690.MR1721891 (2000k:33040)
50. M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant evaluations, Discrete Math. 210 (2000), 151–169.MR1731612 (2001g:33029)
51. M. Schlosser, A nonterminating 8φ7 summation for the root system Cr, J. Comput. Appl. Math. 160 (2003), 283–296.MR2022620
52. L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, Cambridge, 1966.MR0201688 (34:1570)
53. V. P. Spiridonov, Solitons and Coulomb plasmas, similarity reductions and special functions, Special Functions (Hong Kong, China, June 21–25, 1999), World Sci. Publishing, River Edge, NJ, 2000, pp. 324–338.MR1805993 (2001k:37122)
54. V. P. Spiridonov, An elliptic beta integral, Proc. Fifth International Conference on Difference Equations and Applications (Temuco, Chile, January 3–7, 2000), Taylor and Francis, London, 2001, pp. 273–282
55. On the elliptic beta function, Uspekhi Mat. Nauk 56 (2001), no. 1, 181–182
56. English transl., Russian Math. Surveys 56 (2001), no. 1, 185–186.MR1846786 (2003b:33032);MR2016068 (2004j:33024)
57. V. P. Spiridonov, The factorization method, self-similar potentials and quantum algebras, Special Functions-2000: Current Perspective and Future Directions (Tempe, USA, May 29–June 9, 2000), Kluwer, Dordrecht, 2001, pp. 335–364.MR2006294
58. 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, Ukraine, September 25–30, 2000), NATO Sci. Ser. II Math. Phys. Chem., vol. 35, Kluwer, Dordrecht, 2001, pp. 305–313.MR1873579 (2003b:33019)
59. V. P. Spiridonov, Theta hypergeometric series, Asymptotic Combinatorics with Applications to Mathematical Physics (St. Petersburg, Russia, July 9–23, 2001), Kluwer, Dordrecht, 2002, pp. 307–327.MR2000728 (2004j:33012)
60. V. P. Spiridonov, An elliptic incarnation of the Bailey chain, Internat. Math. Res. Notices 2002, no. 37, 1945–1977.MR1918235 (2003j:33055)
61. V. P. Spiridonov, Modularity and complete ellipticity of some multiple series of hypergeometric type, Teoret. Mat. Fiz. 135 (2003), no. 3, 462–477. (Russian)MR1984450 (2004i:33008)
62. V. P. Spiridonov and A. S. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions, Comm. Math. Phys. 210 (2000), 49–83.MR1748170 (2001e:33008)
63. V. P. Spiridonov and A. S. Zhedanov, Classical biorthogonal rational functions on elliptic grids, C. R. Math. Acad. Sci. Soc. R. Can. 22 (2000), no. 2, 70–76.MR1764720 (2001c:33035)
64. V. P. Spiridonov and A. S. Zhedanov, Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids, Special Functions-2000: Current Perspective and Future Directions (Tempe, USA, May 29–June 9, 2000), Kluwer, Dordrecht, 2001, pp. 365–388.MR2006295 (2004j:33028)
65. V. P. Spiridonov and A. S. Zhedanov, To the theory of biorthogonal rational functions, RIMS Kokyuroku 1302 (2003), 172–192.MR1986520
66. V. Tarasov and A. Varchenko, Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque No. 246 (1997), 1–135.MR1646561 (2000b:17021)
67. S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479–502.MR1920282 (2003h:33018)
68. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, Cambridge, 1996.MR1424469 (97k:01072)
69. J. A. Wilson, Hypergeometric series, recurrence relations and some new orthogonal functions, Ph. D. Thesis, Univ. Wisconsin, Madison, WI, 1978; Orthogonal functions from Gram determinants, SIAM J. Math. Anal. 22 (1991), 1147–1155.MR1112071 (92c:33018)
70. A. S. Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory 101 (1999), 303–329.MR1726460 (2000i:33014)