An algebraic interpretation of the multivariate $q$-Krawtchouk polynomials

TL;DR

This paper provides an algebraic framework for understanding multivariate quantum $q$-Krawtchouk polynomials, deriving their key properties through $q$-rotation operators acting on $q$-oscillator states.

Contribution

It introduces an algebraic interpretation of multivariate $q$-Krawtchouk polynomials using $q$-rotations, extending the understanding to multiple variables.

Findings

Derived orthogonality, duality, and recurrence relations.

Established algebraic interpretation via $q$-rotations.

Extended framework to arbitrary variables.

Abstract

The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations and recurrence relations. The extension to an arbitrary number of variables is presented