Abstract
This paper shows a hypercomplex function theory emerging in the representation of paravector-valued monogenic functions over the (m+1)dimensional Euclidean space through a basic set (or basis) of hypercomplex monogenic polynomials. We derive the properties of the arising hypercomplex Cannon function and present an extension of the well-known Whittaker-Cannon theorem to special monogenic functions defined in an open hyperball in Double-struck capital Rm+1. More precisely, we determine what conditions should be applied to a basic set of special monogenic polynomials to attain the effectiveness property in an open hyperball employing Hadamard's three-hyperballs theorem. We also provide a necessary and sufficient condition for a special monogenic Cannon series to represent every function near the origin that is special monogenic there. Additionally, we investigate the effectiveness of a non-Cannon basis and show that the underlying hypercomplex Cannon function maintains similar properties in both cases, the Cannon basis and the non-Cannon basis.