Basic Properties of Non-Stationary Ruijsenaars Functions

Abstract

For any variable number, a non-stationary Ruijsenaars function was recently introduced as a natural generalization of an explicitly known asymptotically free solution of the trigonometric Ruijsenaars model, and it was conjectured that this non-stationary Ruijsenaars function provides an explicit solution of the elliptic Ruijsenaars model. We present alternative series representations of the non-stationary Ruijsenaars functions, and we prove that these series converge. We also introduce novel difference operators called $\mathcal{T}$ which, as we prove in the trigonometric limit and conjecture in the general case, act diagonally on the non-stationary Ruijsenaars functions.

References in corpus (3)

• A direct approach to the bispectral problem for the Ruijsenaars-Macdonald q-difference operators
• Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case
• Non-Stationary Ruijsenaars Functions for $\kappa =t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra

Cited by in corpus (6)

• Construction of eigenfunctions for the elliptic Ruijsenaars difference operators
• Non-stationary difference equation for q-Virasoro conformal blocks
• Non-Stationary Ruijsenaars Functions for $\kappa =t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra
• Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation
• $K$-theoretic wall-crossing formulas and multiple basic hypergeometric series
• Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation