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.