# An analogue of big q-Jacobi polynomials in the algebra of symmetric functions

## TL;DR

This paper constructs a new five-parameter family of bases in the algebra of symmetric functions, extending properties of orthogonal polynomials and linking to infinite-dimensional q-Beta distributions.

## Contribution

It introduces a novel inhomogeneous basis in symmetric functions, utilizing big q-Jacobi polynomials and multivariate interpolation polynomials for infinite variables.

## Key findings

- Bases are orthogonal in weighted Hilbert spaces.
- Embedded algebra of symmetric functions into continuous functions on Omega.
- Connected to infinite-dimensional q-Beta distributions.

## Abstract

The main result of the paper is a construction of a five-parameter family of new bases in the algebra of symmetric functions. These bases are inhomogeneous and share many properties of systems of orthogonal polynomials on an interval of the real line. This means, in particular, that the algebra of symmetric functions is embedded into the algebra of continuous functions on a certain compact space Omega, and under this realization, our bases turn into orthogonal bases of weighted Hilbert spaces corresponding to certain probability measures on Omega. These measures are of independent interest --- they are an infinite-dimensional analogue of the multidimensional q-Beta distributions. Our construction uses the big q-Jacobi polynomials and an extension of the Knop-Okounkov-Sahi multivariate interpolation polynomials to the case of infinite number of variables.