Basic Properties of Non-Stationary Ruijsenaars Functions

TL;DR

This paper introduces new series representations and convergence proofs for non-stationary Ruijsenaars functions, and proposes difference operators that act diagonally on these functions, advancing understanding of elliptic integrable systems.

Contribution

It provides alternative series representations, proves their convergence, and introduces difference operators acting diagonally, enhancing the theoretical framework of non-stationary Ruijsenaars functions.

Findings

Series representations of non-stationary Ruijsenaars functions are convergent.

New difference operators ${\mathcal T}$ are introduced and shown to act diagonally in the trigonometric limit.

Conjectures are made about the operators' actions in the general elliptic case.

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.