Michael Woodbury

Affiliation: University of Cologne

Email: woodbury@math.uni-koeln.de

Title Of Talk: Generalized Frobenius partitions and powers of the Jacobi theta function

Abstract: Let $\vartheta(z;\tau):=\sum_{n\in \mathbb{Z}} e^{\pi i n^2\tau+2\pi i n (z+\frac{1}{2})}$ be the Jacobi theta function. If we let $q:=e^{2\pi i \tau}$ and $\zeta:=e^{2\pi i z}$, then we are interested in computing the $q$-series $F_{k,n}(q)$ in the Fourier expansion $\theta(z;\tau)^k=\sum_{n\in \mathbb{Q}} F_{k,n}(q)\zeta^{n}$. We give a general recursive formula for finding the $q$-series $F_{k,n}(q)$. Of particular interest is $F_{k,\frac{k}{2}}(q)$ because of its connection to certain generalized Frobenius partitions. For example, when $k=1$, up to some normalization factor, $F_{1,\frac{1}{2}}(q)$ is the generating series for the partition function. In the talk, our recursive formula and its connection to generalized Frobenius partitions will be discussed. This is joint work with Kathrin Bringmann and Larry Rolen.

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