Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

TL;DR

This paper extends the concept of moment differentiation to generalized multisums, providing a characterization of multisummability for formal solutions of singularly perturbed moment differential equations, thus broadening the scope of applicable equations.

Contribution

It introduces a new approach to generalized multisummability using integral representations and estimates, applicable to a wider class of singularly perturbed equations.

Findings

Characterization of multisummability for formal solutions

Extension of moment differentiation to generalized multisums

Application to singularly perturbed differential and fractional equations

Abstract

The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.