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Canonical bases for \(\mathfrak {sl}(2,\mathbb {C})\)-modules of spherical monogenics in dimension 3. (English) Zbl 1249.30136

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 [Math. Methods Appl. Sci. 33, No. 4, 394–411 (2010; Zbl 1195.30068)]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable

Citations:

Zbl 1195.30068
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