In the development of a model for my PhD, I've encountered a situation where, if I am able to find an explicit formula to calculate an element of the inverse of a Hankel matrix, this could eventually lead to a very nice simplification of my model.
The matrix is an $n \times n$ Hankel matrix, where $n$ can be any positive integer, and its elements are defined by
$$H_{ij} = \dfrac{x^{i + j - 1}}{i + j - 1},$$ where $x$ is a positive real number.
I have been trying to find ways of approaching this, which I don't even know if it is possible. I do not expect anyone to give me a direct answer, but I was wondering if there is some literature available on how to approach these types of problems, or even to know how I can verify that such expression exists.
I found this paper, which does exactly what I would like to do (but for a different matrix), and I thought about using a similar approach, but I don't have enough mathematics background to even understand what was done in the paper...