Elements of the q-Askey scheme in the algebra of symmetric functions

TL;DR

This paper develops a hierarchy of symmetric functions analogous to the q-Askey scheme, introducing multivariable q-Racah functions and their degenerations, which form orthogonal systems with interesting measures linked by limit transitions.

Contribution

It constructs multivariable q-Racah symmetric functions and explores their degenerations into other orthogonal symmetric functions, expanding the algebraic and analytical framework of the q-Askey scheme.

Findings

Construction of multivariable q-Racah symmetric functions.

Degeneration into big q-Jacobi, q-Meixner, and Al-Salam--Carlitz symmetric functions.

Orthogonality measures linked by limit transitions.

Abstract

The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.