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

TL;DR

This paper generalizes recent findings on the Taylor coefficients of the theta function heta_3 at 1 to the entire heta_3(q) function for any elliptic modulus, using cumulant analysis and proposing new conjectures.

Contribution

It extends previous results on heta_3(1) to heta_3(q) for all elliptic moduli, introducing new methods and conjectures.

Findings

Derived properties of cumulants for heta_3(q)

Generalized congruence conjectures for integer sequences

Extended understanding of theta function Taylor coefficients

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.