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

SIGMA 7 (2011), 101, 54 pages      arXiv:1111.0115      https://doi.org/10.3842/SIGMA.2011.101
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

### A Relativistic Conical Function and its Whittaker Limits

Simon Ruijsenaars
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received April 30, 2011, in final form October 23, 2011; Published online November 01, 2011

Abstract
In previous work we introduced and studied a function $R(a_{+},a_{-},{\bf c};v,\hat{v})$ that is a generalization of the hypergeometric function ${}_2F_1$ and the Askey-Wilson polynomials. When the coupling vector ${\bf c}\in{\mathbb C}^4$ is specialized to $(b,0,0,0)$, $b\in{\mathbb C}$, we obtain a function ${\mathcal R}(a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of ${}_2F_1$ and the $q$-Gegenbauer polynomials. The function ${\mathcal R}$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser system of $A_1$ type, whereas the function $R$ corresponds to $BC_1$, and is the joint eigenfunction of four hyperbolic Askey-Wilson type difference operators. We show that the ${\mathcal R}$-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function ${\mathcal R}$ converges to a joint eigenfunction of the latter four difference operators.

Key words: relativistic Calogero-Moser system; relativistic Toda system; relativistic conical function; relativistic Whittaker function.

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References

1. Ruijsenaars S.N.M., A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type, Comm. Math. Phys. 206 (1999), 639-690.
2. Ruijsenaars S.N.M., A generalized hypergeometric function. II. Asymptotics and $D_4$ symmetry, Comm. Math. Phys. 243 (2003), 389-412.
3. Ruijsenaars S.N.M., A generalized hypergeometric function. III. Associated Hilbert space transform, Comm. Math. Phys. 243 (2003), 413-448.
4. van de Bult F.J., Ruijsenaars' hypergeometric function and the modular double of ${\mathcal U}_q(sl_2({\mathbb C}))$, Adv. Math. 204 (2006), 539-571, math.QA/0501405.
5. Faddeev L., Modular double of a quantum group, in Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., Vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, 149-156, math.QA/9912078.
6. van de Bult F.J., Rains E.M., Stokman J.V., Properties of generalized univariate hypergeometric functions, Comm. Math. Phys. 275 (2007), 37-95, math.CA/0607250.
7. Spiridonov V.P., Classical elliptic hypergeometric functions and their applications, in Elliptic Integrable Systems (2004, Kyoto), Rokko Lect. in Math., Vol. 18, Kobe University, 2005, 253-287, math.CA/0511579.
8. Ruijsenaars S.N.M., Parameter shifts, $D_4$ symmetry, and joint eigenfunctions for commuting Askey-Wilson type difference operators, in Proceedings RAQIS2003, Editors D. Arnaudon, J. Avan, L. Frappat, É. Ragoucy and P. Sorba, J. Phys. A: Math. Gen. 37 (2004), 481-495.
9. Ruijsenaars S.N.M., Quadratic transformations for a function that generalizes ${}_2F_1$ and the Askey-Wilson polynomials, in Askey Festschrift issue (Bexbach 2003 Proceedings), Ramanujan J. 13 (2007), 339-364.
10. Ruijsenaars S.N.M., A relativistic hypergeometric function, in Proceedings OPSFA2003, Editors C. Berg and J. S. Christiansen, J. Comput. Appl. Math. 178 (2005), 393-417.
11. Ruijsenaars S.N.M., Generalized Lamé functions. II. Hyperbolic and trigonometric specializations, J. Math. Phys. 40 (1999), 1627-1663.
12. Ruijsenaars S.N.M., Hilbert space theory for reflectionless relativistic potentials, Publ. Res. Inst. Math. Sci. 36 (2000), 707-753.
13. van Diejen J.F., Integrability of difference Calogero-Moser systems, J. Math. Phys. 35 (1994), 2893-3004.
14. Ruijsenaars S.N.M., Hilbert-Schmidt operators vs. integrable systems of elliptic Calogero-Moser type. I. The eigenfunction identities, Comm. Math. Phys. 286 (2009), 629-657.
15. Komori Y., Noumi M., Shiraishi J., Kernel functions for difference operators of Ruijsenaars type and their applications, SIGMA 5 (2009), 054, 40 pages, arXiv:0812.0279.
16. Digital Library of Mathematical Functions, Release date 2010-05-07, National Institute of Standards and Technology, http://dlmf.nist.gov.
17. Ruijsenaars S.N.M., Relativistic Toda systems, Comm. Math. Phys. 133 (1990), 217-247.
18. Ruijsenaars S.N.M., Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case, Comm. Math. Phys. 115 (1988), 127-165.
19. Kharchev S., Lebedev D., Semenov-Tian-Shansky M., Unitary representations of $U_q(\mathfrak{sl}(2,\mathbb{R}))$, the modular double and the multiparticle $q$-deformed Toda chains, Comm. Math. Phys. 225 (2002), 573-609, hep-th/0102180.
20. Olshanetsky M.A., Rogov V.-B.K., Unitary representations of the quantum Lorentz group and quantum relativistic Toda chain, Theoret. and Math. Phys. 130 (2002), 299-322, math.QA/0110182.
21. Etingof P., Whittaker functions on quantum groups and $q$-deformed Toda operators, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 9-25, math.QA/9901053.
22. Cherednik I., Ma X., A new take on spherical Whittaker and Bessel functions, arXiv:0904.4324.
23. Ruijsenaars S.N.M., Relativistic Lamé functions revisited, in Proceedings Tokyo SIDE IV Conference, Editors J. Hietarinta, F. Nijhoff and J. Satsuma, J. Phys. A: Math. Gen. 34 (2001), 10595-10612.
24. Ruijsenaars S.N.M., Relativistic Lamé functions: completeness vs. polynomial asymptotics, in Papers dedicated to Tom Koornwinder, Indag. Math. (N.S.) 14 (2003), 515-544.
25. Ruijsenaars S.N.M., Finite-dimensional soliton systems, in Integrable and Superintegrable Systems, Editor B. Kupershmidt, World Sci. Publ., Teaneck, NJ, 1990, 165-206.
26. van Diejen J.F., Kirillov A.N., Formulas for $q$-spherical functions using inverse scattering theory of reflectionless Jacobi operators, Comm. Math. Phys. 210 (2000), 335-369.
27. Koornwinder T.H., Jacobi functions as limit cases of $q$-ultraspherical polynomials, J. Math. Anal. Appl. 148 (1990), 44-54.
28. Askey R., Andrews G. E., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
29. Badertscher E., Koornwinder T.H., Continuous Hahn polynomials of differential operator argument and analysis on Riemannian symmetric spaces of constant curvature, Canad. J. Math. 44 (1992), 750-773.
30. Luke Y.L., The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York - London, 1969.
31. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.
32. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.
33. Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
34. Kurokawa N., Multiple sine functions and Selberg zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 61-64.
35. Faddeev L.D., Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249-254, hep-th/9504111.
36. Woronowicz S.L., Quantum exponential function, Rev. Math. Phys. 12 (2000), 873-920.
37. Ruijsenaars S.N.M., A unitary joint eigenfunction transform for the A$\Delta$Os $\exp(ia_{\pm}d/dz)+\exp(2\pi z/a_{\mp})$, in Proceedings Helsinki SIDE VI (2004), Editors J.F. van Diejen and R.G. Halburd, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 253-294.
38. Barnes E.W., The theory of the double gamma function, Lond. Phil. Trans. (A) 196 (1901), 265-387.
39. Stokman J.V., Hyperbolic beta integrals, Adv. Math. 190 (2005), 119-160, math.QA/0303178.
40. van de Bult F.J., Hyperbolic hypergeometric functions, Ph.D. Thesis, University of Amsterdam, 2007.
41. Faddeev L.D., Kashaev R.M., Volkov A.Y., Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality, Comm. Math. Phys. 219 (2001), 199-219, hep-th/0006156.
42. Kashaev R., The non-compact quantum dilogarithm and the Baxter equation, J. Statist. Phys. 102 (2001), 923-936.
43. Ponsot B., Teschner J., Clebsch-Gordan and Racah-Wigner coefficients for continuous series of representations of ${\mathcal U}_q(\mathfrak{sl}(2,{\mathbb R}))$, Comm. Math. Phys. 224 (2001), 613-655, math.QA/0007097.