BC_n-symmetric polynomials

TL;DR

This paper explores two families of BC_n-symmetric polynomials, providing new difference equations, identities, and proofs, and introduces related symmetric functions and integral conjectures for classical symmetric spaces.

Contribution

It offers new difference equations, identities, and proofs for BC_n-symmetric polynomials, along with novel symmetric functions and integral conjectures.

Findings

Derived difference equations for interpolation polynomials

Proved new identities and generalizations of classical hypergeometric transformations

Introduced new symmetric functions and integral conjectures

Abstract

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.