Skip to main content
SpringerLink
Log in

Unit Circle Elliptic Beta Integrals

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Math. Appl. 71, Cambridge Univ. Press, Cambridge, 1999.

  2. E.W. Barnes, “On the theory of the multiple gamma function,” Trans. Cambridge Phil. Soc. 19 (1904), 374–425.

    MATH  Google Scholar 

  3. J.F. van Diejen and V.P. Spiridonov, “An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums,” Math. Res. Letters 7 (2000), 729–746.

    MathSciNet  MATH  Google Scholar 

  4. J.F. van Diejen and V.P. Spiridonov, “Elliptic Selberg integrals,” Internat. Math. Res. Notices, no. 20 (2001), 1083–1110.

  5. L.D. Faddeev, “Discrete Heisenberg-Weyl group and modular group,” Lett. Math. Phys. 34 (1995), 249–254; “Modular double of a quantum group,” Conf. Moshé Flato 1999, vol. I, Math. Phys. Stud. 21, Kluwer, Dordrecht, 2000, pp. 149–156.

  6. L.D. Faddeev, R.M. Kashaev, and A.Yu. Volkov, “Strongly coupled quantum discrete Liouville Theory. I: Algebraic approach and duality,” Commun. Math. Phys. 219 (2001), 199–219.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Felder and A. Varchenko, “The elliptic gamma function and SL(3,ℤ) ⋉ℤ 3,” Adv. Math. 156 (2000), 44–76.

    Article  MathSciNet  MATH  Google Scholar 

  8. R.A. Gustafson, “A generalization of Selberg's beta integral,” Bull. Am. Math. Soc., New Ser. 22 (1990), 97–105.

    Article  MATH  MathSciNet  Google Scholar 

  9. 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.

    Article  MATH  MathSciNet  Google Scholar 

  10. F.H. Jackson, “The basic gamma-function and the elliptic functions,” Proc. Roy. Soc. London A76 (1905), 127–144.

  11. 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.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Kharchev, D. Lebedev, and M. Semenov-Tian-Shansky, “Unitary representations of U q (sl(2,ℝ)), the modular double and the multiparticle q-deformed Toda chains,” Commun. Math. Phys. 225 (2002), 573–609.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Kurokawa, “Multiple sine functions and Selberg zeta functions,” Proc. Japan Acad. 67 A (1991), 61–64.

  14. Yu. Manin, “Lectures on zeta functions and motives (according to Deninger and Kurokawa),” Astérisque 228(4) (1995), 121–163.

    Google Scholar 

  15. M. Nishizawa and K. Ueno, Integral Solutions of Hypergeometric q-Difference Systems with |q| = 1, Physics and Combinatorics (Nagoya, 1999), World Scientific, River Edge, 2001, pp. 273–286.

  16. B. Ponsot and J. Teschner, “Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U q (sl(2,ℝ)),” Commun. Math. Phys. 224 (2001), 613–655.

    Google Scholar 

  17. M. Rahman, “An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions,” Can. J. Math. 38 (1986), 605–618.

    MATH  Google Scholar 

  18. E.M. Rains, Transformations of Elliptic Hypergeometric Integrals, preprint math.QA/0309252.

  19. S.N.M. Ruijsenaars, “First order analytic difference equations and integrable quantum systems,” J. Math. Phys. 38 (1997), 1069–1146.

    Article  MATH  MathSciNet  Google Scholar 

  20. S.N.M. Ruijsenaars, “Generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type,” Commun. Math. Phys. 206 (1999), 639–690.

    Article  MATH  MathSciNet  Google Scholar 

  21. S.N.M. Ruijsenaars, “A generalized hypergeometric function III. Associated Hilbert space transform,” Commun. Math. Phys. 243 (2003), 413–448.

    MATH  MathSciNet  Google Scholar 

  22. T. Shintani, ‘On a Kronecker limit formula for real quadratic field,” J. Fac. Sci. Univ. Tokyo 24 (1977), 167–199.

    MATH  MathSciNet  Google Scholar 

  23. 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; “On the elliptic beta function,” Russ. Math. Surveys 56(1) (2001), 185–186.

    Google Scholar 

  24. V.P. Spiridonov, “Theta hypergeometric integrals,” Algebra i Analiz 15 (2003), 161–215 (St. Petersburg Math. J. 15 (2004), 929–967).

  25. V.P. Spiridonov, “A Bailey tree for integrals,” Theor. Math. Phys. 139 (2004), 536–541.

    Article  MathSciNet  Google Scholar 

  26. J.V. Stokman, “Hyperbolic beta integrals,” Adv. Math. 190 (2004), 119–160.

    MathSciNet  Google Scholar 

  27. Y. Takeyama, “The q-twisted cohomology and the q-hypergeometric function at |q| = 1,” Publ. Res. Inst. Math. Sci. 37 (2001)(1), 71–89.

    Google Scholar 

  28. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1986.

  29. V.P. Spiridonov, “Short proofs of the elliptic beta integrals,” Ramanujan J., to appear (preprint math. CA/0408369).

  30. V.P. Spiridonov, “Elliptic hypergeometric functions,” Habilitation thesis, JINR, Dubna, September 2004 (Russian).

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile

    J. F. Van Diejen

  2. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980, Russia

    V. P. Spiridonov

Authors
  1. J. F. Van Diejen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. P. Spiridonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

2000 Mathematics Subject Classification: Primary—11F50, 11L05, 33D05, 33D67

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van Diejen, J.F., Spiridonov, V.P. Unit Circle Elliptic Beta Integrals. Ramanujan J 10, 187–204 (2005). https://doi.org/10.1007/s11139-005-4846-x

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-005-4846-x

Keywords

Search

Navigation