#### Vol. 11, No. 9, 2017

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On the algebraic structure of iterated integrals of quasimodular forms

### Nils Matthes

Vol. 11 (2017), No. 9, 2113–2130
##### Abstract

We study the algebra ${\mathsc{ℐ}}^{QM}$ of iterated integrals of quasimodular forms for ${SL}_{2}\left(ℤ\right)$, which is the smallest extension of the algebra ${QM}_{\ast }$ of quasimodular forms which is closed under integration. We prove that ${\mathsc{ℐ}}^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the ${M}_{\ast }$-subalgebra ${\mathsc{ℐ}}^{M}$ of ${\mathsc{ℐ}}^{QM}$ of iterated integrals of modular forms.

##### Keywords
quasimodular forms, iterated integrals
Primary: 11F11
Secondary: 11F67