### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 037, 11 pages      arXiv:1701.08960      https://doi.org/10.3842/SIGMA.2017.037
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### Gustafson-Rakha-Type Elliptic Hypergeometric Series

Hjalmar Rosengren
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden

Received February 02, 2017, in final form May 29, 2017; Published online June 01, 2017

Abstract
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.

Key words: elliptic hypergeometric series; multivariable hypergeometric series; Jackson summation; Bailey transformation.

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References

1. Bhatnagar G., $D_n$ basic hypergeometric series, Ramanujan J. 3 (1999), 175-203.
2. Bhatnagar G., Schlosser M., $C_n$ and $D_n$ very-well-poised ${}_{10}\phi_9$ transformations, Constr. Approx. 14 (1998), 531-567.
3. Coskun H., Gustafson R.A., Well-poised Macdonald functions $W_\lambda$ and Jackson coefficients $\omega_\lambda$ on $BC_n$, in Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, Amer. Math. Soc., Providence, RI, 2006, 127-155, math.CO/0412153.
4. Date E., Jimbo M., Kuniba A., Miwa T., Okado M., Exactly solvable SOS models. II. Proof of the star-triangle relation and combinatorial identities, in Conformal Field Theory and Solvable Lattice Models (Kyoto, 1986), Adv. Stud. Pure Math., Vol. 16, Academic Press, Boston, MA, 1988, 17-122.
5. Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, in The Arnold-Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, 171-204.
6. Gustafson R.A., Some $q$-beta and Mellin-Barnes integrals with many parameters associated to the classical groups, SIAM J. Math. Anal. 23 (1992), 525-551.
7. Gustafson R.A., Rakha M.A., $q$-beta integrals and multivariate basic hypergeometric series associated to root systems of type $A_m$, Ann. Comb. 4 (2000), 347-373.
8. Milne S.C., Newcomb J.W., ${\rm U}(n)$ very-well-poised $_{10}\phi_9$ transformations, J. Comput. Appl. Math. 68 (1996), 239-285.
9. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, math.CO/0402113.
10. Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. of Math. 171 (2010), 169-243, math.QA/0309252.
11. Rosengren H., A proof of a multivariable elliptic summation formula conjectured by Warnaar, in $q$-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), Contemp. Math., Vol. 291, Amer. Math. Soc., Providence, RI, 2001, 193-202, math.CA/0101073.
12. Rosengren H., Elliptic hypergeometric series on root systems, Adv. Math. 181 (2004), 417-447, math.CA/0207046.
13. Rosengren H., Schlosser M., Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations, Indag. Math. 14 (2003), 483-513, math.CA/0304249.
14. Rosengren H., Schlosser M., Multidimensional matrix inversions and elliptic hypergeometric series on root systems, in preparation.
15. Schlosser M., A new multivariable $_6\psi_6$ summation formula, Ramanujan J. 17 (2008), 305-319, math.CA/0607122.
16. Spiridonov V.P., Theta hypergeometric integrals, St. Petersburg Math. J. 15 (2003), 929-967, math.CA/0303205.
17. Spiridonov V.P., Vartanov G.S., Elliptic hypergeometry of supersymmetric dualities, Comm. Math. Phys. 304 (2011), 797-874, arXiv:0910.5944.
18. Spiridonov V.P., Warnaar S.O., New multiple $_6\psi_6$ summation formulas and related conjectures, Ramanujan J. 25 (2011), 319-342.
19. van Diejen J.F., Spiridonov V.P., Elliptic Selberg integrals, Int. Math. Res. Not. 2001 (2001), 1083-1110.
20. Warnaar S.O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502, math.QA/0001006.
21. Whittaker E.T., Watson G.N., A course of modern analysis, 4th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.