# A note on Fox's H function in the light of Braaksma's results

## TL;DR

This paper refines the power series expansion of Fox's H function near its positive singularity by correcting previous parameter restrictions and providing a simpler proof based on Braaksma's theorem.

## Contribution

It corrects and simplifies the expansion of Fox's H function around its singularity, extending previous results with a more precise convergence analysis.

## Key findings

- Provided a corrected parameter restriction for the expansion.
- Simplified the proof using a generalization of Braaksma's theorem.
- Identified the abscissa of convergence of the inverse factorial series.

## Abstract

In our previous works we found a power series expansion of a particular case of Fox's $H$ function $H^{q,0}_{p,q}$ in a neighborhood of its positive singularity. An inverse factorial series expansion of the integrand of $H^{q,0}_{p,q}$ served as our main tool. However, a necessary restriction on parameters is missing in those works. In this note we fill this gap and give a simpler and shorter proof of the expansion around the positive singular point. We further identify more precisely the abscissa of convergence of the underlying inverse factorial series. Our new proof hinges on a slight generalization of a particular case of Braaksma's theorem about analytic continuation of Fox's $H$ function.