Classical orthogonal polynomials revisited

TL;DR

This paper revisits the algebraic theory of classical orthogonal polynomials, providing new characterizations and proving their uniqueness among polynomial families through algebraic and affine transformations.

Contribution

It offers a cyclical proof of key characterizations and introduces two new characterizations, establishing the uniqueness of Hermite, Laguerre, Jacobi, and Bessel polynomials.

Findings

Provided a cyclical proof of classical polynomial characterizations

Introduced two new characterizations of classical orthogonal polynomials

Proved the uniqueness of the four classical families up to affine transformations

Abstract

This manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel polynomials. It is presented a cyclical proof of some of the most relevant characterizations, particularly those due to Al-Salam and Chihara, Bochner, Hahn, Maroni, and McCarthy. Two apparently new characterizations are also added. Moreover, it is proved through an equivalence relation that, up to constant factors and affine changes of variables, the four families of polynomials named above are the only families of classical orthogonal polynomials.