Summation formulae for elliptic hypergeometric series

Warnaar, S.

doi:10.1090/S0002-9939-04-07558-6

Summation formulae for elliptic hypergeometric series
HTML articles powered by AMS MathViewer

by S. Ole Warnaar PDF

Proc. Amer. Math. Soc. 133 (2005), 519-527 Request permission

Abstract:

Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.

References

George E. Andrews, On $q$-analogues of the Watson and Whipple summations, SIAM J. Math. Anal. 7 (1976), no. 3, 332–336. MR 399529, DOI 10.1137/0507026
George Andrews and Alexander Berkovich, The WP-Bailey tree and its implications, J. London Math. Soc. (2) 66 (2002), no. 3, 529–549. MR 1934290, DOI 10.1112/S0024610702003617
J. F. van Diejen and V. P. Spiridonov, An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), no. 5-6, 729–746. MR 1809297, DOI 10.4310/MRL.2000.v7.n6.a6
J. F. van Diejen and V. P. Spiridonov, Elliptic Selberg integrals, Internat. Math. Res. Notices 20 (2001), 1083–1110. MR 1857597, DOI 10.1155/S1073792801000526
J. F. Van Diejen and V. P. Spiridonov, Modular hypergeometric residue sums of elliptic Selberg integrals, Lett. Math. Phys. 58 (2001), no. 3, 223–238 (2002). MR 1892922, DOI 10.1023/A:1014567500292
J. F. van Diejen and V. P. Spiridonov, Elliptic beta integrals and modular hypergeometric sums: an overview, Rocky Mountain J. Math. 32 (2002), no. 2, 639–656. Conference on Special Functions (Tempe, AZ, 2000). MR 1934909, DOI 10.1216/rmjm/1030539690
Igor B. Frenkel and Vladimir 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. MR 1429892
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
Y. Kajihara and M. Noumi, Multiple elliptic hypergeometric series. An approach from the Cauchy determinant, Indag. Math. (N.S.) 14 (2004), 395–421.
E. Koelink, Y. van Norden and H. Rosengren, Elliptic U(2) quantum group and elliptic hypergeometric series, Commun. Math. Phys. 245 (2004), 519–537.
B. Nassrallah and Mizan Rahman, On the $q$-analogues of some transformations of nearly-poised hypergeometric series, Trans. Amer. Math. Soc. 268 (1981), no. 1, 211–229. MR 628455, DOI 10.1090/S0002-9947-1981-0628455-0
Mizan Rahman and Arun Verma, Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335 (1993), no. 1, 277–302. MR 1074149, DOI 10.1090/S0002-9947-1993-1074149-8
E. M. Rains, Transformations of elliptic hypergometric integrals, arXiv:math.QA/0309252.
Hjalmar 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. MR 1874532, DOI 10.1090/conm/291/04903
H. Rosengren, Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417–447.
H. Rosengren and M. Schlosser, Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations, Indag. Math. (N.S.) 14 (2004), 483–513.
H. Rosengren and M. Schlosser, On Warnaar’s elliptic matrix inversion and Karlsson–Minton-type elliptic hypergeometric series, arXiv:math.CA/0309358.
Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
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, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 35, Kluwer Acad. Publ., Dordrecht, 2001, pp. 305–313. MR 1873579
V. P. Spiridonov, Theta hypergeometric series, in Asymptotic Combinatorics with Applications to Mathematical Physics, pp. 307–327, V. A. Malyshev and A. M. Vershik, eds., (Kluwer Academic Publishers, Dordrecht, 2002).
V. P. Spiridonov, An elliptic incarnation of the Bailey chain, Int. Math. Res. Not. 37 (2002), 1945–1977. MR 1918235, DOI 10.1155/S1073792802205127
V. P. Spiridonov, Theta hypergeometric integrals, Algebra i Analiz 15 (2003), 161–215.
Vyacheslav Spiridonov and Alexei Zhedanov, Classical biorthogonal rational functions on elliptic grids, C. R. Math. Acad. Sci. Soc. R. Can. 22 (2000), no. 2, 70–76 (English, with English and French summaries). MR 1764720
S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), no. 4, 479–502. MR 1920282, DOI 10.1007/s00365-002-0501-6
S. O. Warnaar, Extensions of the well-poised and elliptic well-poised Bailey lemma, Indag. Math. (N.S.) 14 (2004), 571–588.