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

We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $θ_{3}\left(1\right)$ to the case $θ_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are obtained by carefully studying the properties of the cumulants associated to a $θ_{3}$ (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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