Expansion of solutions of higher-order operator equations into generalized Hartogs series

Abstract

The paper studies solutions of the higher-order operator equation Lᵐu=0 for a pair of linear operators satisfying the commutation relation LM-ML=I. A unique Almansi-type decomposition is proved for such solutions, with an explicit formula for its components. On this basis, functions that are polyanalytic in one variable and holomorphic in another are expanded into generalized Hartogs series with polyanalytic coefficients. Since the coefficients are not holomorphic, the classical Hartogs lemma cannot be applied directly, and the domain of convergence is therefore determined through the components. It is proved that the upper regularization of the function describing the radius of convergence is subharmonic, that the domain of convergence is a pseudoconvex Hartogs domain, and that the function extends to this domain. The results are illustrated by an explicit example and figures.