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


SIGMA 16 (2020), 105, 26 pages      arXiv:2006.07171      https://doi.org/10.3842/SIGMA.2020.105
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Basic Properties of Non-Stationary Ruijsenaars Functions

Edwin Langmann a, Masatoshi Noumi b and Junichi Shiraishi c
a) Physics Department, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden
b) Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
c) Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received June 15, 2020, in final form October 08, 2020; Published online October 21, 2020

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 T which, as we prove in the trigonometric limit and conjecture in the general case, act diagonally on the non-stationary Ruijsenaars functions.

Key words: elliptic integrable systems; elliptic hypergeometric functions; Ruijsenaars systems.

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