Abstract
We give elementary proofs of the univariate elliptic beta integral with bases |q|,|p| < 1 and its multiparameter generalizations to integrals on the A n and C n 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.
Similar content being viewed by others
References
Andrews, G.E., Askey, R., Roy, R.: Special functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge Univ. Press, Cambridge (1999)
Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319) (1985)
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)
Felder, G., Varchenko, A.: The elliptic gamma function and \(SL(3,{\mathbb Z})\ltimes {\mathbb Z}^3\). Adv. in Math. 156, 44–76 (2000)
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)
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)
Rahman, M.: An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions. Can. J. Math. 38, 605–618 (1986)
Rains, E.M.: Transformations of elliptic hypergeometric integrals. preprint (2003), math.QA/0309252
Ruijsenaars, S. N. M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997)
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)
Spiridonov, V.P.: Warnaar, S.O.: Inversions of integral operators and elliptic beta integrals on root systems. Adv. in Math. 207, 91–132 (2006)
Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108, 575–633 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Richard Askey on the occasion of his seventieth birthday.
2000 Mathematics Subject Classification Primary—33D99, 33E05
This work is supported in part by the Russian Foundation for Basic Research (RFBR) grant no. 03-01-00781.
Rights and permissions
About this article
Cite this article
Spiridonov, V.P. Short proofs of the elliptic beta integrals. Ramanujan J 13, 265–283 (2007). https://doi.org/10.1007/s11139-006-0252-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11139-006-0252-2