Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

Abstract

AbstractWe prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ h 1 , $$h_2$$ h 2 of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ f = h 1 - x ¯ h 2 . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.