Complete Orthogonal Appell Systems for Spherical Monogenics

Abstract

In this paper, we investigate properties of Gelfand-Tsetlin bases mainly for spherical monogenics, that is, for spinor valued or Clifford algebra valued homogeneous solutions of the Dirac equation in the Euclidean space.Recently it has been observed that in dimension 3 these bases form an Appell system. We show that Gelfand-Tsetlin bases of spherical monogenics form complete orthogonal Appell systems in any dimension. Moreover, we study the corresponding Taylor series expansions for monogenic functions. We obtain analogous results for spherical harmonics as well.

Generalized Taylor series in hermitian Clifford analysis

“…Note that in [38] another GT-basis for the same Bergman space B 2 (B 2n ; C) has been obtained, using (real) one step branching, breaking the symmetry from SO(m) to SO(m − 1), with m = 2n.…”

“…[24,21,22,23] in a direct analytic way starting from spherical harmonics, and in e.g. [47,37,4,38,40,39] by the so-called Gel'fand-Tsetlin [GT] approach. The notion of GT-basis stems from group representation theory: every irreducible finite dimensional module over a classical Lie group has its GT-basis (see e.g.…”

A Matrix Recurrence for Systems of Clifford Algebra-Valued Orthogonal Polynomials

“…Finally we notice that several authors ( [16,23]) prefer to consider monogenic functions as solutions of the Dirac equation…”

“…The results in [3] were later generalized in a systematic way in the paper [23] where the author constructed Gelfand-Tsetlin bases for Clifford-valued homogeneous solutions of the Dirac operator in arbitrary dimensional Euclidean space.…”

“…Theorem 2.1 permits to describe a recursive method for building monogenic polynomial bases in arbitrary dimensions starting from a basis in R 2 with values in the Clifford algebra C 0,2 (see [23,3]). Indeed, an orthogonal basis for the space M k (R n+1 , C 0,n ) is formed by the polynomials…”