Romanovski polynomials in selected physics problems

TL;DR

This paper reviews the properties of Romanovski polynomials, a lesser-known family of orthogonal polynomials, highlighting their applications in various physics problems and presenting new insights into their orthogonality and Rodrigues formula.

Contribution

It provides a comprehensive review of Romanovski polynomials, including new observations on their orthogonality and developments related to their Rodrigues formula.

Findings

Romanovski polynomials exhibit finite orthogonality.

They are applicable in quantum mechanics, quark physics, and random matrix theory.

New properties of their orthogonality are identified.

Abstract

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.