A new generalisation of Macdonald polynomials

TL;DR

This paper introduces a new family of symmetric polynomials that generalize Macdonald polynomials and connect to previously known inhomogeneous functions, with explicit matrix product evaluations.

Contribution

It presents a novel family of symmetric polynomials depending on four parameters, extending Macdonald polynomials and linking to Borodin's inhomogeneous functions.

Findings

Polynomials reduce to Macdonald polynomials at u=v=0

Polynomials recover Borodin's functions at q=0, u=v=s

Explicit matrix product formula derived for the polynomials

Abstract

We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters $(q,t)$ and polynomial in a further two parameters $(u,v)$. We evaluate these polynomials explicitly as a matrix product. At $u=v=0$ they reduce to Macdonald polynomials, while at $q=0$, $u=v=s$ they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.