Basics on growth orders of polymonogenic functions

Abstract

In this paper, we introduce generalizations of the classical growth order and the growth type of analytic functions in the context of polymonogenic functions. Polymonogenic functions are null-solutions of higher integer order iterates of a generalized higher dimensional Cauchy-Riemann operator. One of the main goals is to prove generalizations of the famous Lindelöf-Pringsheim theorem linking explicitly these growth orders and growth types with the Taylor series coefficients in the context of this function class.

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

“…Similar to Theorem from De Almeida and Kraußhar, we obtain for this class of functions a generalization of the Lindelöf‐Pringsheim theorem in the following version.…”

“…The proof can be done in the same steps as the one given in Theorem from De Almeida and Kraußhar incorporating some of the technical details that arise in the consideration of the larger function class that we address here. Basically, the new features arise from the application of the projection formula developed in Subsection 3.2; after that, the proof follows along the same lines as in the polymonogenic setting given in Theorem from De Almeida and Kraußhar …”

“…A further serious problem that we solved recently in De Almeida and Kraußhar was to study whether there are explicit links between the growth orders and the Taylor coefficients in the context of the higher‐order systems for k >1. In contrast to the monogenic case treated in Constales et al, it turned out that one does not obtain a direct analogue of the classical Lindelöf‐Pringsheim formula for k >1 in terms of an equality relation between the growth order and the Taylor coefficients.…”

“…In this paper, we address the most general setting of polynomial Cauchy‐Riemann equations with constant complex coefficients of the form where we consider arbitrarily many eigenvalues λ i with arbitrary multiplicities k i . In this context, we have to combine the toolkit that we developed to address polymonogenic Cauchy‐Riemann equations in De Almeida and Kraußhar and that to address eigenfunctions of Cauchy‐Riemann operators. Therefore, we first look at the simpler case for k >1 where λ may be an arbitrarily chosen complex number.…”

On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices

“…Let F(X) be an entire matrix function of positive order ρ and type τ, and let b ≠ 0 be the real constant. en, the matrix function F ⋆ (bX + A) is an entire matrix function of the growth order ρ and the growth type τ|b| ρ [26].…”