Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials
Tewodros Amdeberhan, Leonid G. Fel, Ken Ono
TL;DR
The paper constructs a hierarchy of quasimodular forms by applying Ramanujan’s generating function $\Omega(X)$ to build $Y_n(q)$ of weight $n$ on $\Gamma_0(4)$ from the weight-1 theta form $\theta(q)^2$ and $F(X)=e^{X/2}$, with symmetric avatars that match the polynomials $T_n(\mathbf{x}^k)$ arising from numerical semigroups. It establishes a precise link between these quasimodular forms and symmetric function theory via partition Eisenstein traces, and proves two conjectures of Fel by showing $\widetilde{Y}_n(\mathbf{x}) = T_n(\mathbf{x})$ and connecting $T_n$ to $f_n$ under a sign-alternating substitution. The results illuminate deep connections between quasimodular forms, umbral calculus, and the topology of spin manifolds through the $\widehat{A}$-genus, while providing explicit closed forms and identities for symmetric polynomials tied to semigroup syzygies. Overall, the work bridges modular-type objects with combinatorial and topological invariants, offering new avenues for generating and understanding structured families of symmetric polynomials.
Abstract
Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ 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 $F(X)$. Using the weight 1 form $θ(q)^2$ and $F(X)=\exp(X/2)$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $Γ_0(4)$ whose symmetric function avatars $\widetilde{Y}_n(\pmb{x}^k)$ are the symmetric polynomials $T_n(\pmb{x}^k)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n(\pmb{x}^k).$ Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\widehat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes.