A basic triad in Macdonald theory

TL;DR

This paper explores the deep connections between Macdonald polynomials, Baker-Akhiezer functions, and Noumi-Shiraishi series, revealing they form a closely related triad with different definitions but interconnected properties.

Contribution

It demonstrates that the Noumi-Shiraishi series reduces to Baker-Akhiezer functions at specific parameters, establishing a new link among these objects.

Findings

The NS series reduces to BA functions at t=q^{-m}.

Macdonald polynomials, BA functions, and NS series form a closely tied triad.

BA functions provide simple linear equations for analysis.

Abstract

Within the context of wavefunctions of integrable many-body systems, rational multivariable Baker-Akhiezer (BA) functions were introduced by O. Chalykh, M. Feigin and A. Veselov and, in the case of the trigonometric Ruijsenaars-Schneider system, can be associated with a reduction of the Macdonald symmetric polynomials at $t=q^{-m}$ with integer partition labels substituted by arbitrary complex numbers. A parallel attempt to describe wavefunctions of the bispectral trigonometric Ruijsenaars-Schneider problem was made by M. Noumi and J. Shiraishi who proposed a power series that reduces to the Macdonald polynomials at particular values of parameters. It turns out that this power series also reduces to the BA functions at $t=q^{-m}$, as we demonstrate in this letter. This makes the Macdonald polynomials, the BA functions and the Noumi-Shiraishi (NS) series a closely tied {\it triad} of objects, which have very different definitions, but are straightforwardly related with each other. In particular, theory of the BA functions provides a nice system of simple linear equations, while the NS functions provide a nice way to represent the multivariable BA function explicitly with arbitrary number of variables.