The Radon–Penrose Transformation for Quaternionic k-Regular Functions on Right-Type Groups

Abstract

The right-type groups are nilpotent Lie groups of step two having a pair of anticommutative operators, and many aspects of quaternionic analysis can be generalized to this kind of groups. In this paper, we use the twistor transformation to study the tangential k-Cauchy–Fueter equations and quaternionic k-regular functions on these groups. We introduce the twistor space over the \((4n+r)\)-dimensional complex right-type groups and use twistor transformation to construct an explicit Radon–Penrose type integral formula to solve the holomorphic tangential k-Cauchy–Fueter equation on these groups. When restricted to the real right-type group, this formula provides solutions to tangential k-Cauchy–Fueter equations. In particular, it gives us many k-regular polynomials.