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

TL;DR

This paper explores the properties and applications of complementary Romanovski-Routh polynomials, revealing their connections to orthogonal polynomials on the unit circle, Whittaker functions, and Coulomb wave functions, with implications for quantum mechanics.

Contribution

It establishes new links between CRR-polynomials, orthogonal polynomials on the unit circle, and special functions like Whittaker and Coulomb wave functions.

Findings

CRR-polynomials are related to orthogonal polynomials on the unit circle.

They are connected to a subfamily of Whittaker functions including Coulomb wave functions.

An electrostatic interpretation of the zeros of CRR-polynomials is provided.

Abstract

We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-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 are related to a special class of orthogonal polynomials on the unit circle. As another main result, we have established their connection to a class of functions which are related to a subfamily of Whittaker functions that includes those associated with the Bessel functions and the regular Coulomb wave functions. An electrostatic interpretation for the zeros of CRR-polynomials is also considered.