$q$-Rotations and Krawtchouk polynomials

TL;DR

This paper provides an algebraic framework for quantum $q$-Krawtchouk polynomials using $ ext{U}_q(sl_2)$ and $q$-oscillators, deriving their properties and extending results to affine $q$-Krawtchouk polynomials.

Contribution

It introduces an algebraic interpretation of quantum $q$-Krawtchouk polynomials via $ ext{U}_q(sl_2)$ and extends the framework to affine $q$-Krawtchouk polynomials.

Findings

Derived orthogonality, generating functions, and recurrence relations.

Expressed polynomials as matrix elements of $q$-rotation operators.

Extended results to affine $q$-Krawtchouk polynomials.

Abstract

An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown to arise as matrix elements of unitary "$q$-rotation" operators expressed as $q$-exponentials in the $\mathcal{U}_{q}(sl_{2})$ generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine $q$-Krawtchouk polynomials, through a duality relation.