  This paper show that two doubly infinite series generate a totally
nonnegative Hurwitz-type matrix if and only if their ratio represents an
S-functions of a certain kind. The doubly infinite case needs a specific
approach, since the ratios have no correspondent Stieltjes continued fraction.
Another forthcoming publication (see Dyachenko, arXiv:1608.04440) offers a
shorter improved version of this result as well as its application to the
Hurwitz stability. Nevertheless, the proof presented here illustrates features
of totally nonnegative Hurwitz-type matrices better.
