The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states

TL;DR

This paper interprets multivariate Krawtchouk polynomials as matrix elements of rotation group representations in oscillator models, providing a group-theoretic framework that generalizes to higher dimensions.

Contribution

It introduces a novel algebraic interpretation of multivariate Krawtchouk polynomials via SO(d+1) group representations, extending previous models.

Findings

Polynomials arise as matrix elements of rotation group representations.

Properties are derived using group-theoretic methods.

Generalization to higher dimensions is achieved.

Abstract

An algebraic interpretation of the bivariate Krawtchouk polynomials is provided in the framework of the 3-dimensional isotropic harmonic oscillator model. These polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in 3 dimensions. Many of their properties are derived by exploiting the group-theoretic setting. The bivariate Tratnik polynomials of Krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters. It is explained how the approach generalizes naturally to (d+1) dimensions and allows to interpret multivariate Krawtchouk polynomials as matrix elements of SO(d+1) unitary representations. Indications are given on the connection with other algebraic models for these polynomials.