A Generalized Jacobi Theta Function and Quasimodular Forms

Abstract

In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let ${\tilde M_*}({\Gamma _1})$ denote the graded ring of quasi-modular forms on the full modular group Γ= PSL(2, ℤ). This is the ring generated by G₂, G₄, G₆, and graded by assigning to each G _k the weight where ${G_k} = - \frac{{{B_k}}}{{2k}} + \sum\limits_{n = 1}^\infty {\left( {{{\sum\limits_{d|n} d }^{k - 1}}} \right)} {q^n}\left( {k \geqslant 2,{B_k} = kth Bernoulli number} \right)$ are the classical Eisenstein series, all of which except G ₂ are modular.