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

SIGMA 14 (2018), 019, 69 pages      arXiv:1408.0305      https://doi.org/10.3842/SIGMA.2018.019
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

Multivariate Quadratic Transformations and the Interpolation Kernel

Eric M. Rains
Department of Mathematics, California Institute of Technology, USA

Received September 12, 2017, in final form February 27, 2018; Published online March 08, 2018

Abstract
We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the ''interpolation kernel'', an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.

Key words: quadratic transformations; elliptic special functions.

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References

  1. Betea D., Wheeler M., Zinn-Justin P., Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures, J. Algebraic Combin. 42 (2015), 555-603, arXiv:1405.7035.
  2. de Bruijn N.G., On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.) 19 (1955), 133-151.
  3. van de Bult F.J., An elliptic hypergeometric integral with $W(F_4)$ symmetry, Ramanujan J. 25 (2011), 1-20, arXiv:0909.4793.
  4. van de Bult F.J., Two multivariate quadratic transformations of elliptic hypergeometric integrals, arXiv:1109.1123.
  5. van de Bult F.J., Elliptic hypergeometric functions, in Symmetries and Integrability of Difference Equations, CRM Ser. Math. Phys., Springer, Cham, 2017, 43-74.
  6. van de Bult F.J., Rains E.M., Limits of multivariate elliptic hypergeometric biorthogonal functions, arXiv:1110.1458.
  7. Cooper S., The Askey-Wilson operator and the ${}_6\Phi_5$ summation formula, South East Asian J. Math. Math. Sci. 1 (2002), 71-82.
  8. Derkachev S.E., Spiridonov V.P., The Yang-Baxter equation, parameter permutations, and the elliptic beta integral, Russian Math. Surveys 68 (2013), 1027-1072, arXiv:1205.3520.
  9. van Diejen J.F., Integrability of difference Calogero-Moser systems, J. Math. Phys. 35 (1994), 2983-3004.
  10. van Diejen J.F., Spiridonov V.P., An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), 729-746.
  11. van Diejen J.F., Spiridonov V.P., Elliptic Selberg integrals, Internat. Math. Res. Notices 2001 (2001), 1083-1110.
  12. Filali G., Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end, J. Geom. Phys. 61 (2011), 1789-1796, arXiv:1012.0516.
  13. 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.
  14. Gustafson R.A., A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97-105.
  15. Ismail M., Rains E.M., Stanton D., Orthogonality of very well-poised series, in preparation.
  16. Izergin A.G., Partition function of a six-vertex model in a finite volume, Dokl. Akad. Nauk SSSR 297 (1987), 331-333.
  17. Kawanaka N., A $q$-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999), 157-176.
  18. Koornwinder T.H., Askey-Wilson polynomials for root systems of type $BC$, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204.
  19. Korepin V.E., Calculation of norms of Bethe wave functions, Comm. Math. Phys. 86 (1982), 391-418.
  20. Langer R., Schlosser M.J., Warnaar S.O., Theta functions, elliptic hypergeometric series, and Kawanaka's Macdonald polynomial conjecture, SIGMA 5 (2009), 055, 20 pages, arXiv:0905.4033.
  21. Okada S., Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (1998), 337-367.
  22. Okounkov A., ${\rm BC}$-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), 181-207, q-alg/9611011.
  23. Rains E.M., Ruijsenaars S., Difference operators of Sklyanin and van Diejen type, Comm. Math. Phys. 320 (2013), 851-889, arXiv:1203.0042.
  24. Rains E.M., ${\rm BC}_n$-symmetric polynomials, Transform. Groups 10 (2005), 63-132, math.QA/0112035.
  25. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, math.CO/0402113.
  26. Rains E.M., A difference-integral representation of Koornwinder polynomials, in Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, Amer. Math. Soc., Providence, RI, 2006, 319-333, math.CA/0409437.
  27. Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. of Math. 171 (2010), 169-243, math.QA/0309252.
  28. Rains E.M., Elliptic Littlewood identities, J. Combin. Theory Ser. A 119 (2012), 1558-1609, arXiv:0806.0871.
  29. Rains E.M., Vazirani M., Vanishing integrals of Macdonald and Koornwinder polynomials, Transform. Groups 12 (2007), 725-759.
  30. Rains E.M., Elliptic double affine Hecke algebras, arXiv:1709.02989.
  31. Spiridonov V.P., On the elliptic beta function, Russian Math. Surveys 56 (2001), 185-186.
  32. Spiridonov V.P., A Bailey tree for integrals, Theoret. and Math. Phys. 139 (2004), 536-541, math.CA/0312502.
  33. Spiridonov V.P., Warnaar S.O., Inversions of integral operators and elliptic beta integrals on root systems, Adv. Math. 207 (2006), 91-132, math.CA/0411044.
  34. Tate J., A review of non-Archimedean elliptic functions, in Elliptic Curves, Modular Forms, & Fermat's Last Theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, 162-184.
  35. Warnaar S.O., Bisymmetric functions, Macdonald polynomials and $\mathfrak{sl}_3$ basic hypergeometric series, Compos. Math. 144 (2008), 271-303, math.CO/0511333.
  36. Zinn-Justin P., Sum rule for the eight-vertex model on its combinatorial line, in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat., Vol. 40, Springer, Heidelberg, 2013, 599-637, arXiv:1202.4420.

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