An explicit formula for the inverse of a factorial Hankel matrix

TL;DR

This paper derives an explicit formula for the inverse of a factorial Hankel matrix, proving its invertibility for all sizes and providing a concrete computational method.

Contribution

It introduces a closed-form expression for the inverse of a specific factorial Hankel matrix, a novel result in matrix analysis.

Findings

The factorial Hankel matrix is invertible for all sizes.

An explicit formula for the inverse matrix is provided.

The inverse can be computed directly using the derived formula.

Abstract

We consider the $n\times n$ Hankel matrix $H$ whose entries are defined by $H_{ij}=1/s_{i+j}$ where $s_k=(k-1)!$ and prove that $H$ is invertible for all $n\in\mathbb{N}$ by providing an explicit formula for its inverse matrix.