Biorthogonal Expansion of Non-Symmetric Jack Functions

TL;DR

This paper develops a biorthogonal expansion for the Cayley transform of non-symmetric Jack functions using non-symmetric Jack polynomials and Meixner-Pollaczek polynomials, via Cherednik-Opdam transform calculations.

Contribution

It introduces a novel biorthogonal expansion framework for non-symmetric Jack functions involving explicit polynomial coefficients.

Findings

Derived explicit coefficients for the expansion.

Connected the expansion to Cherednik-Opdam transform.

Enhanced understanding of non-symmetric Jack functions' structure.

Abstract

We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function.