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