Branching rules for symmetric Macdonald polynomials and sl_n basic hypergeometric series

TL;DR

This paper introduces a new generalization of symmetric Macdonald polynomials, deriving branching rules, identities, and hypergeometric series formulas, advancing the understanding of their algebraic and combinatorial properties.

Contribution

It presents a novel one-parameter family of polynomials with new Pieri, symmetry, and summation formulas, enriching the theory of symmetric functions and hypergeometric series.

Findings

Established Pieri formula and evaluation symmetry

Derived a new multiple q-Gauss summation formula

Developed q-difference equations for the generalized polynomials

Abstract

A one-parameter generalisation R_{\lambda}(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for R_{\lambda}(X;b). We also prove a new multiple q-Gauss summation formula and several further results for sl_n basic hypergeometric series based on R_{\lambda}(X;b).