On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials

TL;DR

This paper explores the orthogonality properties of Atkin-like polynomials and reveals their unexpected connection to generalized Faber polynomials and extremal quasimodular forms, providing new insights into their structure and relationships.

Contribution

It establishes a novel link between Atkin-like polynomials, generalized Faber polynomials, and extremal quasimodular forms through orthogonal polynomial expansions.

Findings

Orthogonal polynomial expansion coefficients relate to Fourier coefficients of extremal quasimodular forms.

Atkin-like polynomials are connected to generalized Faber polynomials via orthogonal polynomial theory.

New properties of Atkin-like polynomials are clarified in the context of modular forms.

Abstract

In this paper, we consider the Atkin-like polynomials that appeared in the study of normalized extremal quasimodular forms of depth 1 on $SL_{2}(\mathbb{Z})$ by Kaneko and Koike as orthogonal polynomials and clarify their properties. By considering Atkin-like polynomials in terms of orthogonal polynomials, we prove an unexpected connection between generalized Faber polynomials, which are closely related to certain bases of the vector space of weakly holomorphic modular forms, and normalized extremal quasimodular forms. In particular, we show that the orthogonal polynomial expansion coefficients of the generalized Faber polynomials by the Atkin-like polynomials appear in the Fourier coefficients of normalized extremal quasimodular forms multiplied by certain (weakly) holomorphic modular forms.