Keywords

• hypercomplex analysis
• octonionic function theory
• monogenic function
• Weierstrass function
• Bergman space
• Hardy space
• Cayley-Dickson algebra

Abstract In this talk we present a series of new results in octonionic monogenic function theory. We introduce generalizations of the Weierstrass $\wp$ and $\zeta$ function associated with eight-dimensional lattices that have an octonionic multiplication and explain some connections to some possible relations and applications to Class Field Theory.

Furthermore, we also give some explicit applications of these kind of functions to the study of Bergman and Hardy spaces in the octonionic cases. Octonionic monogenic generalizations of the cotangent and the cosecant can be obtained as subseries of the octonionic Weierstrass $\wp$ functions. These functions turn out to be the building blocks for the reproducing octonionic Bergman and Szegö kernel of strip domains in $\mathbb{R}^8$.

Summarizing, these new functions seem to play a key role in octonionic function theories and their applications to number theory and function spaces.