  Some extensions of the (third-class) Romanovski polynomials (also called
Romanovski/pseudo-Jacobi polynomials), which appear in bound-state
wavefunctions of rationally-extended Scarf II and Rosen-Morse I potentials, are
considered. For the former potentials, the generalized polynomials satisfy a
finite orthogonality relation, while for the latter an infinite set of
relations among polynomials with degree-dependent parameters is obtained. Both
types of relations are counterparts of those known for conventional
polynomials. In the absence of any direct information on the zeros of the
Romanovski polynomials present in denominators, the regularity of the
constructed potentials is checked by taking advantage of the disconjugacy
properties of second-order differential equations of Schr\"odinger type. It is
also shown that on going from Scarf I to Scarf II or from Rosen-Morse II to
Rosen-Morse I potentials, the variety of rational extensions is narrowed down
from types I, II, and III to type III only.
