On the Atkin Polynomials

TL;DR

This paper identifies Atkin polynomials as associated Jacobi polynomials, leveraging orthogonal polynomial theory to reveal new properties and demonstrate the potential of co-recursive polynomials to generate interesting orthogonal sets.

Contribution

It provides the first explicit identification of Atkin polynomials with associated Jacobi polynomials and explores their properties using orthogonal polynomial theory.

Findings

Atkin polynomials are expressed in terms of associated Jacobi polynomials.

New properties of Atkin polynomials are established through asymptotic analysis.

Co-recursive polynomials can generate interesting orthogonal polynomial sets.

Abstract

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials. This shows that co-recursive polynomials may lead to interesting sets of orthogonal polynomials.