Rational generalized Stieltjes functions

The rational meromorphic functions on $\mathbb{C}\backslash\mathbb{R}$ are studied. We consider the some classes of one, as the generalized Nevanlinna $\mathbf{N}_{\kappa}$ and generalized Stieltjes $\mathbf{N}_{\kappa}^{k}$ classes. By Euclidean algorithm, we can find indices $\kappa$ and $k$, i.e. determine which class the function belongs to $\mathbf{N}_{\kappa}^{k}$.