On a class of formal power series and their summability

TL;DR

This paper characterizes a class of formal power series with recursive coefficients through their summability properties, linking them to differential equations and extending results to q-Gevrey, q-difference, and moment-differential equations.

Contribution

It introduces a new characterization of formal power series via summability and extends the analysis to q-Gevrey, q-difference, and moment-differential equations.

Findings

Characterization of formal power series with recursive coefficients via summability.

Identification of differential equations with formal solutions as these series.

Extension of results to q-Gevrey, q-difference, and moment-differential equations.

Abstract

A family of formal power series, such that its coefficients satisfy a recursion formula, is characterized in terms of the summability, in the sense of J. P. Ramis, of its elements along certain well chosen directions. We describe a set of ordinary differential equations which have, as a formal solution, one of such formal power series, and also describe the actual solutions which own such formal power series as Gevrey asymptotic expansion. The main results are extended into the framework of $q-$Gevrey series and $q-$difference equations, and also to the case of moment-differential equations, whose formal solutions have their coefficients' growth governed by a so-called strongly regular sequence, appearing as a generalization of the Gevrey-like sequences.