Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

Abstract

Ramanujan derived a sequence of even weight 2n quasimodular forms $U_{2n}\left(q\right)$ from derivatives of Jacobi's weight 3/2 theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F\left(X\right)$. Using the weight 1 form $\theta {\left(q\right)}^2$ and $F\left(X\right)=\exp (X/2)$, we obtain a sequence $\left\{Y_n\right(q\left)\right\}$ of weight n quasimodular forms on ${\Gamma }_0\left(4\right)$ whose symmetric function avatars ${\tilde{Y}}_n\left(x^k\right)$ are the symmetric polynomials $T_n\left(x^k\right)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n\left(x^k\right)$. Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\hat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes.