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We consider properties and applications of a sequence of polynomials known ascomplementary Romanovski-Routh polynomials (CRR polynomials for short). Thesepolynomials, which follow from the Romanovski-Routh polynomials or complexifiedJacobi polynomials, are known to be useful objects in the studies of theone-dimensional Schr\"{o}dinger equation and also the wave functions of quarks.One of the main results of this paper is to show how the CRR-polynomials arerelated to a special class of orthogonal polynomials on the unit circle. Asanother main result, we have established their connection to a class offunctions which are related to a subfamily of Whittaker functions that includesthose associated with the Bessel functions and the regular Coulomb wavefunctions. An electrostatic interpretation for the zeros of CRR-polynomials isalso considered.

Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions