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

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