On the solutions to linear systems of moment differential equations with variable coefficients
Alberto Lastra
TL;DR
This work establishes local existence and analyticity for linear moment-differential systems with variable coefficients, extending prior constant-coefficient results by analyzing $\partial_m y(z)=A(z)y(z)+b(z)$ on discs around the origin. It leverages strongly regular sequences and generalized summability to control the growth of solutions and uses majorant (and, in the Puiseux setting, Caputo-type) techniques to obtain holomorphic solutions under Assumptions (A) and (B). The paper also clarifies how first-order moment systems can be transformed into higher-order moment differential equations when a cyclic vector exists, providing conditions under which a companion equation with polynomial coefficients can be derived. By connecting moment differentiation with Caputo fractional calculus, the results unify the moment framework with classical fractional-differential approaches and delineate the domains of analyticity for solutions. These findings offer constructive methods for solving variable-coefficient moment systems and broaden the applicability of moment-differential methods in complex-domain problems.
Abstract
The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is stablished, comparing classical results with the general situation of moment differentiation.