Hypermonogenic functions and Möbius transformations

LetCln be the (universal) Clifford algebra generated bye1, …,en satisfyingeiej+ejei=−2σij,i, j=1, …,n. The Dirac operator inCln is defined by {ie67-1}, wheree0=1. The second author started to develop the theory of functionsf=Σi=0nfiei satisfying the modified Cauchy-Riemann systemxnDf+(n −1)fn=0, calledH-solutions. The power function xm, x=x0+x1e1+…+xnen, in contrast to classical Clifford analysis, is anH-solution. We study an extension ofH-solutions named hypermonogenic functions. They are solutions of the equationxnDf+(n − 1)Q′f=0, where denotes the main involution of the Clifford algebra andQf is given by the decompositionf (x)=Pf (x)+Qf (x)en withPf (x) ,Qf (x) ∈Cln−1. Note that the values of hypermonogenic functions are in the full Clifford algebraCln. Hypermonogenic functions form a rightClnn−1-module. They are closely related to hyperbolic geometry. Locally any hypermonogenic function may be represented as {ie67-2} in terms of some hyperbolic harmonic functionH with values inCln−1. We prove that compositions of hypermonogenic functions with Möbius transformations lead to hypermonogenic functions and discuss related matters.