Quaternionic Hyperbolic Function Theory

We are studying hyperbolic function theory in the skew-field of quaternions. This theory is connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$\displaystyle ds_{k}^{2}=\frac {dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{k}} $$ in the upper half space \(\mathbb {R}_{+}^{4}=\{ \left ( x_{0},x_{1},x_{2},x_{3}\right )\in \mathbb {R}^4 : x_{3}>0\} \). In the case k = 2, the metric is the hyperbolic metric of the Poincare upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function \(x^{m}\,(m\in \mathbb {Z})\), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. We find fundamental k-hyperbolic harmonic functions depending only on the hyperbolic distance and x3. Using these functions we are able to verify a Cauchy type integral formula. Earlier these results have been verified for quaternionic functions depending only on reduced variables \(\left ( x_{0},x_{1},x_{2}\right )\). Our functions are depending on four variables.