# 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.