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

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. Using them, we show that the normalized extremal quasimodular forms have a certain expression by the linear functional corresponding to the Atkin inner product and 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 reveal 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.