Exercise(***). Use the mapping properties of Stieltjes continued fractions to prove that their interlacing, simple and symmetric zeros and poles lie at the origin and the imaginary axes and that the properties and rationality characterize the continued fractions.

Exercise(**). Prove that the continued fractions 've the representation $\beta _{a}(z)=z(\xi _{\infty }+\sum _{k}{\frac {\xi _{k}}{z^{2}+\theta _{k}^{2}}}),{\mbox{ where }}\xi _{\infty },\xi _{k}{\mbox{ and }}\theta _{k},k\in \mathbb {N}$, 're non-negative real numbers, and the fractions 're characterized by it.

The function $\beta _{a}$ is determined by the pre-image of unity (i.e. n points, counting multiplicities), since

Then, the coefficients $a_{k}$ of the continued fraction are the pivots in the Gauss-Jordan elimination algorithm of the following $n\times n$ square Hurwitz matrix: