On the Atkin polynomials

Ahmad El-Guindy and Mourad E. H. Ismail

Vol. 272 (2014), No. 1, 111–130

DOI: 10.2140/pjm.2014.272.111

Abstract

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

Keywords

associated Jacobi polynomials, asymptotics, weight functions, scaled invariant, modular function, birth and death process polynomials, supersingular polynomials, Hecke operators

Mathematical Subject Classification 2010

Primary: 33C47, 11F03

Secondary: 33C05

Milestones

Received: 14 April 2013

Revised: 9 September 2013

Accepted: 16 September 2013

Published: 9 October 2014