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Hypermonogenic functions. (English) Zbl 0965.30020
Ryan, John (ed.) et al., Clifford algebras and their applications in mathematical physics. Papers of the 5th international conference, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999. Volume 2: Clifford analysis. Boston, MA: Birkhäuser. Prog. Phys. 19, 287-302 (2000).
One possible way to get a function theory in higher dimensions is to use the Clifford algebra \(Cl_n\) and to study solutions of \(D_nf=0\), where \(D_n=\sum_{k=0}^n e_k\partial_{x_k}\) is the Dirac operator. These solutions are called left monogenic functions. In comparison with classical function theory and their well-known approach in higher dimensions there appear disadvantages as e.g. the fact that the power function \(x^m\) is not left monogenic, where \(x=\sum^n_{k=0}x_ke_k\). For this reason the authors introduce a modified Dirac operator \[ M_n:=D_N +{n-1\over x_n}Q_{n-1}' \] and call \(f\) a (left) hypermonogenic function if \(M_nf=0\) in some open subset \(\Omega\setminus \{x:x_n =0\} \subset \mathbb{R}^{n+1}\). If a hypermonogenic function \(f=\sum^n_{l=0} f_le_l\) with real functions \(f_l\), then \(f\) is called a \(H^n\)-solution. The authors study properties of hypermonogenic functions, \(H_n\)-solutions, respectively. Especially, all power functions \(x^m\) are \(H_n\)-solutions. It would be interesting to understand how this fact can be used to create a function-theoretical approach for hypermonogenic functions or to complete a function theory for monogenic functions.
For the entire collection see [Zbl 0945.00020].

30G35 Functions of hypercomplex variables and generalized variables