Connections between Romanovski and other polynomials

TL;DR

This paper establishes a mathematical connection between Romanovski polynomials and polynomials solving specific quantum mechanical equations, providing new formulas and relations to classical polynomials.

Contribution

It introduces a natural reworking of the Rodrigues formula to connect Romanovski polynomials with solutions of Schrödinger equations involving Rosen-Morse and Scarf potentials.

Findings

Derived a closed-form generating function

Established recursion relations and addition theorems

Linked Romanovski polynomials to classical polynomials

Abstract

A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schr\"odinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.