A lemma by Kronecker states that the series $f(z):= \sum_{i=0}^{\infty} c_i z^i$ represents a rational function if and only if for every $m \gg 0$ the determinant of the following matrix is equal to zero:

\begin{bmatrix} c_0 & c_1 & \cdots & c_m \\ c_1 & c_2 & \cdots & c_{m+1}\\ \vdots & \vdots & & \vdots \\ c_m & c_{m+1} & \cdots & c_{2m} \end{bmatrix}

Question: What are alternative ways to detect this property, that is $f(z)$ being a rational function?

A lemma by Kronecker states that the series $f(z):= \sum_{i=0}^{\infty} c_i z^i$ represents a rational function if and only if for every $m \gg 0$ the determinant of the following matrix is equal to zero:

\begin{bmatrix} c_0 & c_1 & \cdots & c_m \\ c_1 & c_2 & \cdots & c_{m+1}\\ \vdots & \vdots & & \vdots \\ c_m & c_{m+1} & \cdots & c_{2m} \end{bmatrix}

Question: What are alternative ways to detect this property, that is $f(z)$ being a rational function?

Added later: I am looking for algebraic/analytic characterisations, as I don't have the precise values of $c_i$ at hand. However, I know that $c_i$ are integers if that is of any help.