General Integral Formulas for k-hyper-mono-genic Functions

Abstract

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.