An algebraic characterization of the Kronecker function

We prove that every formal Laurent series in two variables which satisfies the Fay identity belongs to a five-parameter family of functions constructed from the Kronecker function. We use this result to characterize the generating series of extended period polynomials of normalized Hecke eigenforms for $\operatorname{PSL}_2(\mathbb Z)$ studied by Zagier in terms of the period relations and the existence of a suitable factorization.