A general class of n-particle difference Calogero-Moser systems with elliptic potentials is introduced. Besides the step size and two periods, the Hamiltonian depends on nine coupling constants. We prove the quantum integrability of the model for n = 2 and present partial results for n 2 3. In degenerate cases (rational, hyperbolic, or trigonometric limit), the integrability follows for arbitrary particle number from previous work connected with the multivariable q-polynomials of Koornwinder and Macdonald. Liouville integrability of the corresponding classical systems follows as a corollary. Limit transitions lead to various well-known models such as the nonrelativistic Calogero-Moser systems associated with classical root systems and the relativistic Calogero-Moser system.

Abstract. We present an elliptic Macdonald-Morris constant term conjecture in the form of an evaluation formula for a Selberg-type multiple beta integral composed of elliptic gamma functions. By multivariate residue calculus, a summation formula recently conjectured by Warnaar for a multiple modular (or elliptic) hypergeometric series is recovered. When the imaginary part of the modular parameter tends to +∞, our elliptic Macdonald-Morris conjecture follows from a Selberg-type multivariate Nassrallah-Rahman integral due to Gustafson. As a consequence we arrive at a proof for the basic hypergeometric degeneration of Warnaar's sum, which amounts to a multidimensional generalization of Jackson's very-well-poised balanced terminating 8 Φ 7 summation formula. By exploiting its modular properties, the validity of Warnaar's sum at the elliptic level is moreover verified independently for low orders in log(q) (viz. up to order 10).

Limits of a recently introduced n-particle difference Calogero-Moser system with elliptic potentials are studied. We obtain hyperbolic and rational difference Calogero-Moser systems with an eight-parameter external field and (finite) difference Toda chains with four-parameter potentials acting on the boundary particles. Hamiltonians for a number of known integrable n-particle systems, such as Ruijsenaars' relativistic Calogero-Moser and Toda models and their generalizations associated with classical root systems, can be seen as special cases of the Hamiltonians considered in this paper. 0 1995 American Institute of Physics.

Abstract. We present some elliptic beta integrals with a base parameter on the unit circle, together with their basic degenerations.

Abstract. We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials.