Elliptic hyperlogarithms

Let $\mathcal{E}$ be a complex elliptic curve and $S$ be a non-empty finite subset of $\mathcal{E}$. We show that the functions $\tilde{\mathrm{\Gamma }}$ introduced in arXiv:1712.07089 out of string theory motivations give rise to a basis of the minimal algebra $A_{\mathcal{E}\setminus S}$ of holomorphic multivalued functions on $\mathcal{E}\setminus S$ which is stable under integration, introduced in arXiv:2212.03119; this basis is alternative to the basis of $A_{\mathcal{E}\setminus S}$ constructed in loc. cit. using elliptic analogues of the hyperlogarithm functions.