Dealing with the confluent Heun equation, something unimportant to me at the beginning got me curious lately: the Poincaré rank of an irregular singularity.
In particular, that one in the confluent Heun equation at the point $\infty$ is $1$.
I have tried to find the definition of this rank, but its origin seems to be in a paper written in German, which I can't read. Also I have found different definitions in other texts that depend on the context I guess.
So my question is, for the people working in the field of ODEs and special functions, is there a standard/generally accepted definition of the Poincaré rank?
Just for the record, the one that I was assuming to be 'correct' is the following:
We say that the function $F(z)$ has rank $r$ at an irregular singularity $z_i$ (for $z_i$ a complex value, not $\infty$) if it can be written as $$(z-z_i)^{-r-1}F_0(z),$$ where $F_0(z)$ has a convergent Taylor series expansion in some neighborhood containing $z_i$.
If $F$ has rank $r\geq 1$ at $z=\infty$, there exists $F_0(z)$ such that $$F(z)=z^{r-1}F_0(z),$$ where $F_0(z)$ has a convergent Taylor series expansion in $z^{-1}$ around $\infty$ and $F_0(\infty)\neq 0$.
The following discussion can be found in the book Heun's Differential Equations edited by A. Ronveaux.
Suppose that $z=\infty$ is an irregular singularity of the ODE \begin{equation} y^{\prime\prime}(z)+p(z)y^{\prime}(z)+q(z)y(z)=0 \end{equation} with $p$ and $q$ rational functions and that, as $z\to\infty$, \begin{equation} p(z)=\mathcal{O}\left(z^{k_1}\right),\quad q(z)=\mathcal{O}\left(z^{k_2}\right). \end{equation}
Since $p$ and $q$ are rational functions, $k_1$ and $k_2$ are integers, and in the case of one of them being identically zero, we take the corresponding $k_i$ as $-\infty$. Defining \begin{equation} g:=\max\left(k_1,\frac{k_2}{2}\right), \end{equation} the Poincaré rank is defined by \begin{equation} h:=g+1. \end{equation}