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Gelfand-Tsetlin bases for spherical monogenics in dimension 3. (English) Zbl 1253.30056

Summary: The main aim of this paper is to recall the notion of Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac and the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT construction, we obtain explicit orthogonal bases for spherical monogenics in dimension 3. We compare them with those constructed by the first and the second author recently (by a direct analytic approach) and we show in addition that the GT basis has the Appell property with respect to all three variables. The last fact is quite important for future applications.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
31C99 Generalizations of potential theory
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References:

[1] Abul-Ez, M. A. and Constales, D.: On the order of basic series representing Clifford valued functions. Appl. Math. Comput. 142 (2003), no. 2-3, 575-584. · Zbl 1058.30040 · doi:10.1016/S0096-3003(02)00350-8
[2] Andrews, G. E., Askey, R. and Roy, R.: Special functions. Encyclopedia of Mathematics and its applications 71, Cambridge University Press, Cambridge, 1999.
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