Wiman‐Valiron theory for higher dimensional polynomial Cauchy‐Riemann equations

Abstract

In this paper, we introduce different kinds of growth orders for the set of entire solutions to the most general framework of higher‐dimensional polynomial Cauchy‐Riemann equations ∏i=1pfalse(scriptD−λifalse)kif=0, where scriptD:=∂f∂x0+∑i=1nei∂f∂xi is the hypercomplex Cauchy‐Riemann operator, λi are arbitrarily chosen nonzero complex constants, and ki are arbitrarily chosen positive integers. The core ingredient is a projection formula that establishes a relation to the ki‐monogenic component functions, which are null‐solutions to iterates of the Cauchy‐Riemann operator that we studied in earlier works. Furthermore, we briefly outline the analogies of the Lindelöf‐Pringsheim theorem in this context.