On linear systems of moment differential equations with singularities of the first kind
Alberto Lastra, Cruz Prisuelos-Arribas, Victor Soto-Larrosa
TL;DR
This work develops a framework for linear moment-differential systems with a simple pole at the origin, in the form $z\partial_m y=(zA+B)y$, under a good-spectrum condition on $B$. It extends the moment-derivative to $z^\nu$-weighted series, constructs formal Floquet-type solutions, and provides convergence criteria, including a refined hypothesis to handle non-diagonalizable $B$. A central contribution is the generalized matrix exponential $z_m^B$, enabling the explicit assembly of fundamental solutions, with detailed analyses for planar diagonal and Jordan blocks and a systematic extension to arbitrary $B$ under Hypothesis (H3). The results yield analytic solutions in many generalized summability contexts (e.g., $q$-difference, Caputo fractional) and offer a robust toolkit for singularity analysis of first-kind systems, with potential applications in confluent hypergeometric-type problems and beyond.
Abstract
The solution to systems of moment differential equations of the form $z\partial_my=(zA+B)y$ are provided, for a matrix $B$ with general good spectrum. Existence and convergence of Floquet-type solutions is studied. A generalized definition of $z^B$ is given, as a tool to solve the main problem whenever $A\equiv 0$. The theory is illustrated with examples which are important in applications.