Cauchy type integral formulas for k-hypermonogenic functions

Presenter: Sirkka-Liisa Eriksson, Tampere University of Technology, Department of Mathematics, Finland

Joint work with Heikki Orelma

Our aim is to build a function theory based on hyperbolic metric of the Poincare upper half space model and Clifford numbers. We consider harmonic functions with respect to the Laplace-Beltrami operator of the Riemannian metric $ds^{2}=x_{2}^{-% \frac{2k}{n-1}}\left( \sum_{i=0}^{n}dx_{i}^{2}\right) $ and their function theory in $\mathbb{R}^{n+1}$. H. Leutwiler in 1992 discovered that the power function, calculated using the Clifford product, is a conjugate gradient of a harmonic function with respect to the hyperbolic metric. He started to research these type of functions, called H-solutions, satisfying a modified Dirac equation, connected to the hyperbolic metric. All usual trigonometric and exponential functions have a natural extensions to H-solutions. S.-L. Eriksson and H. Leutwiler, extended H-solutions to total algebra valued functions, called hypermonogenic functions. We study generalized hypermonogenic functions, called $k-$hypermonogenic functions. For example the function $\left\vert x\right\vert ^{k-n+1}x^{-1}$ is $k$-hypermonogenic. Note that $0$-hypermonogenic are monogenic and $n-1$ -hypermonogenic functions are hypermonogenic We verify the Cauchy type integral formulas for $k$-hypermonogenic functions where the kernels are calculated using the hyperbolic distance of the Poincare upper half space model. There formulas are different in even and odd cases. Earlier these results have been proved for hypermonogenic functions. The coauthor of the work is Heikki Orelma, Tampere University of Technology.