For alpha ∈ R, the class of alpha-order spherical harmonic functions in an open set O ⊆ S n-1, Halpha (O) is defined as the C2-solutions of Deltaalphau = 0; where Deltaalpha = Deltas + alpha(n + alpha - 2) is the spherical Laplace-Beltrami operator of order alpha and Deltas is the radially independent part of the Laplace operator (see [La1], [So]). In the first part of this thesis, we obtain a Green's integral formula for the functions in H alpha(O) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form, which is slightly different from the Euclidean space. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented. Moreover, we show some applications to our formulas, such as the Painleve's uniqueness theorem; the Weierstruss uniform convergence theorem; the mean value theorem, and Laplace radical expansions etc.; On one hand, our results above are extensions, in the sense of the degrees of operators, to the Cauchy formula of functions in MarW , the class of spherical right monogenic functions of order alpha (see [La1]). On the other hand, they can be regarded as the analogs on the unit sphere to the integral representations to null solutions of iterated Dirac operators in Euclidean space (see [R1], [R3]). In short, our work fills the gap between the these two cases. In order to investigate the connections between spherical monogenic (spherical harmonic) functions on S n-1 and monogenic (harmonic) functions in Euclidean space Rn -1, the Cayley transformation is applied to the Cauchy-Green type formulae (see [R1]) in Rn -1. We obtain another version of Green type formulae on Sn-1. This method also allows us to extend our results to some different type of iterated spherical Dirac operators. To close our thesis, we address our future work with regarding to some applications of our theorems.
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