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Keywords:
spherical monogenics; orthogonal basis; Legendre polynomials; $\mathfrak{sl}(2,{\mathbb{C}})$-module
spherical monogenics; orthogonal basis; Legendre polynomials; $\mathfrak{sl}(2,{\mathbb{C}})$-module
Summary:
Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as ${\mathfrak{sl}}(2,{\mathbb{C}})$-modules. As finite-dimensional irreducible ${\mathfrak{sl}}(2,{\mathbb{C}})$-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. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
References:
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