Combinatorics Seminar: The Taylor coefficients of the Jacobi theta constant $\theta _3$

Seminar | October 8 | 12-1 p.m. | 939 Evans Hall | Note change in date

 Dan Romik, UC Davis

 Department of Mathematics

We study the Taylor expansion around the point $x=1$ of a classical modular form, the Jacobi theta constant $\theta_3$. This leads naturally to a new sequence $(d(n))^\infty_{n=0} =1,1,−1,51,849,−26199,\dots$ of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of $\theta_3$. We prove several results about the numbers $d(n)$ and conjecture that they satisfy the congruence $d(n)\cong (−1)^{n−1} (mod\, 5)$ and other similar congruence relations.