Limits of elliptic hypergeometric integrals. (English) Zbl 1178.33021
Summary: In Ann. Math. (2) 171, No. 1, 169–243 (2010; Zbl 1209.33014), the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (\(q\)-hypergeometric) integral identities as limits from the elliptic level.
MSC:
| 33D70 | Other basic hypergeometric functions and integrals in several variables |
| 33C70 | Other hypergeometric functions and integrals in several variables |
Citations:
Zbl 1209.33014References:
| [1] | Al-Salam, W.A., Ismail, M.E.H.: A q-beta integral on the unit circle and some biorthogonal rational functions. Proc. Am. Math. Soc. 121(2), 553–561 (1994) · Zbl 0835.33011 |
| [2] | Andrews, G.E., Askey, R., Roy, R.: Special functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999) · Zbl 0920.33001 |
| [3] | Dixon, A.L.: On a generalisation of Legendre’s formula \(KE'-(K-E)K'=\frac{1}{2}\pi\) . Proc. Lond. Math. Soc. (2) 3, 206–224 (1905) · JFM 36.0506.01 · doi:10.1112/plms/s2-3.1.206 |
| [4] | Frobenius, G.: Über die elliptischen Functionen zweiter Art. J. Reine Angew. Math. 93, 53–68 (1882) · JFM 14.0389.01 · doi:10.1515/crll.1882.93.53 |
| [5] | Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 96. Cambridge University Press, Cambridge (2004) · Zbl 1129.33005 |
| [6] | Gustafson, R.A.: Some q-beta and Mellin-Barnes integrals with many parameters associated to the classical groups. SIAM J. Math. Anal. 23(2), 525–551 (1992) · Zbl 0764.33008 · doi:10.1137/0523026 |
| [7] | Koornwinder, T.H.: Jacobi functions as limit cases of q-ultraspherical polynomials. J. Math. Anal. Appl. 148(1), 44–54 (1990) · Zbl 0713.33010 · doi:10.1016/0022-247X(90)90026-C |
| [8] | Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995) · Zbl 0824.05059 |
| [9] | Narukawa, A.: The modular properties and the integral representations of the multiple elliptic gamma functions. Adv. Math. 189(2), 247–267 (2004) · Zbl 1077.33024 · doi:10.1016/j.aim.2003.11.009 |
| [10] | Nassrallah, B.: Mizan Rahman. Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials. SIAM J. Math. Anal. 16(1), 186–197 (1985) · Zbl 0564.33009 · doi:10.1137/0516014 |
| [11] | Olver, F.W.J.: Asymptotics and Special Functions. Academic, New York (1974) · Zbl 0303.41035 |
| [12] | Rains, E.M.: Recurrences of elliptic hypergeometric integrals. In: Noumi, M., Takasaki, K. (eds.) Elliptic Integrable Systems. Rokko Lectures in Mathematics, vol. 18, pp. 183–199 (2005). arXiv:math.CA/0504285 |
| [13] | Rains, E.M.: Transformations of elliptic hypergeometric integrals. Ann. Math. (2008, to appear) · Zbl 1209.33014 |
| [14] | Rosengren, H., Schlosser, M.: Elliptic determinant evaluations and the Macdonald identities for affine root systems. Compos. Math. 142(4), 937–961 (2006) · Zbl 1104.15009 · doi:10.1112/S0010437X0600203X |
| [15] | Ruijsenaars, S.N.M.: A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type. Commun. Math. Phys. 206(3), 639–690 (1999) · Zbl 0944.33014 · doi:10.1007/PL00005522 |
| [16] | Selberg, A.: Remarks on a multiple integral. Norsk Mat. Tidsskr. 26, 71–78 (1944) · Zbl 0063.06870 |
| [17] | Spiridonov, V.P.: Classical elliptic hypergeometric functions and their applications. In: Noumi, M., Takasaki, K. (eds.) Elliptic Integrable Systems. Rokko Lectures in Mathematics, vol. 18, pp. 253–287 (2005). arXiv:math/0511579 |
| [18] | Stokman, J.V.: Generalized Cherednik-Macdonald identities. arXiv:0708.0934 · Zbl 1173.33313 |
| [19] | Stokman, J.V.: Hyperbolic beta integrals. Adv. Math. 190(1), 119–160 (2005) · Zbl 1072.33012 · doi:10.1016/j.aim.2003.12.003 |
| [20] | van de Bult, F., Rains, E.M., Stokman, J.: Properties of generalized univariate hypergeometric functions. Comm. Math. Phys. 276(1), 37–95 (2007) · Zbl 1144.33007 |
| [21] | van Diejen, J.F., Spiridonov, V.P.: Elliptic Selberg integrals. Int. Math. Res. Not. 20, 1083–1110 (2001) · Zbl 1010.33010 |
| [22] | van Diejen, J.F., Spiridonov, V.P.: Unit circle elliptic beta integrals. Ramanujan J. 10(2), 187–204 (2005) · Zbl 1160.33309 · doi:10.1007/s11139-005-4846-x |
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.
