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arXiv · 1909.01508

Taylor coefficients of the Jacobi $\theta_{3}\left( q \right)$ function

Tanay Wakhare·Christophe Vignat
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Abstract

We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $\theta_{3}\left(1\right)$ to the case $\theta_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are obtained by carefully studying the properties of the cumulants associated to a $\theta_{3}$ (or discrete normal) distributed random variable. This article also states some congruence conjectures about integers sequences that generalize the one studied by D. Romik.

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Tanay Wakhare, Christophe Vignat. 2019-09-04. Taylor coefficients of the Jacobi $\theta_{3}\left( q \right)$ function. https://arxiv.org/abs/1909.01508

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