The Ramanujan Journal

, Volume 13, Issue 1–3, pp 265–283 | Cite as

Short proofs of the elliptic beta integrals



We give elementary proofs of the univariate elliptic beta integral with bases |q|,|p| < 1 and its multiparameter generalizations to integrals on the An and Cn root systems. We prove also some new unit circle multiple elliptic beta integrals, which are well defined for |q| = 1, and their p → 0 degenerations.


Beta integrals Elliptic hypergeometric functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  2. 2.
    Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319) (1985)Google Scholar
  3. 3.
    van Dicjcn, J.F., Spiridonov, V.P.: An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums. Math. Res. Letters 7, 729–746 (2000); Elliptic Selberg integrals. Internat. Math. Res. Notices 20, 1083–1110 (2001); Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58, 223–238 (2001); Unit circle elliptic beta integrals. Ramanujan J. 10, 187–204 (2005)Google Scholar
  4. 4.
    Felder, G., Varchenko, A.: The elliptic gamma function and \(SL(3,{\mathbb Z})\ltimes {\mathbb Z}^3\). Adv. in Math. 156, 44–76 (2000)Google Scholar
  5. 5.
    Gustafson, R.A.: Some q-beta and Mellin-Barnes integrals with many parameters associated to classical groups. SIAM J. Math. Anal. 23, 525–551 (1992); Some q-beta integrals on SU(n) and Sp(n) that generalize the Askey-Wilson and Nassrallah-Rahman integrals. SIAM J. Math. Anal. 25, 441–449 (1994); Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras. Trans. Amer. Math. Soc. 341, 69–119 (1994)Google Scholar
  6. 6.
    Kharchev, S., Lebedev, D., Semenov-Tian-Shansky, M.: Unitary representations of U q(sl(2,ℝ)), the modular double and the multiparticle q-deformed Toda chains. Commun. Math. Phys. 225, 573–609 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Rahman, M.: An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions. Can. J. Math. 38, 605–618 (1986)Google Scholar
  8. 8.
    Rains, E.M.: Transformations of elliptic hypergeometric integrals. preprint (2003), math.QA/0309252Google Scholar
  9. 9.
    Ruijsenaars, S. N. M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Spiridonov, V.P.: An elliptic beta integral. In: Proc. Fifth ICDEA (Temuco, Chile, January 2–7, 2000), Taylor and Francis, London (2001 pp. 273–282); On the elliptic beta function. Russ. Math. Surveys 56, 185–186 (2001); Theta hypergeometric integrals, Algebra i Analiz 15, 161–215 (2003) (St. Petersburg Math. J. 15, 929–967 (2004)); A Bailey tree for integrals. Theor. Math. Phys. 139, 536–541 (2004)Google Scholar
  11. 11.
    Spiridonov, V.P.: Warnaar, S.O.: Inversions of integral operators and elliptic beta integrals on root systems. Adv. in Math. 207, 91–132 (2006)CrossRefGoogle Scholar
  12. 12.
    Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108, 575–633 (1992)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Bogoliubov Laboratory of Theoretical PhysicsJINR, DubnaMoscowRussia

Personalised recommendations