Canonical bases for sl(2,C)-modules of spherical monogenics in dimension 3

TL;DR

This paper explores the structure of spherical monogenic modules in three dimensions, revealing their canonical orthogonal bases form an Appell system and relate to Legendre polynomials, providing explicit expressions.

Contribution

It demonstrates that the orthogonal bases of these modules form an Appell system and coincide with recent constructions, offering explicit formulas involving Legendre polynomials.

Findings

Bases form an Appell system

Bases coincide with recent constructions

Explicit expressions involve Legendre polynomials

Abstract

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as sl(2,C)-modules. As finite-dimensional irreducible sl(2,C)-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Guerlebeck. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.