Generalizations of Fueter’s theorem

Abstract

In relation to the solution of the Vekua system for axial type monogenic functions, generalizations of Fueter's Theorem are discussed. We show that if f is a holomorphic function in one complex variable, then for any underlying space R n 1 the induced function k+(n-1)/2 f (x 0 +x)P k (x), where P k (x) is left-monogenic and homogeneous of degree k, is left-monogenic whenever k +(n-1)/2 is a non-negative integer. If the space dimension n + 1 is odd, then the above also holds for k being non-negative integers.