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

Tanay Wakhare; Christophe Vignat

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

Number Theory 2020-03-18 v2

Authors: Tanay Wakhare , Christophe Vignat

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.

Keywords

jacobi theta function taylor coefficients cumulants elliptic modulus congruence conjectures discrete normal distribution

Cite

@article{arxiv.1909.01508,
  title  = {Taylor coefficients of the Jacobi $\theta_{3}\left( q \right)$ function},
  author = {Tanay Wakhare and Christophe Vignat},
  journal= {arXiv preprint arXiv:1909.01508},
  year   = {2020}
}

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17 pages, second version

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

Abstract

Keywords

Cite

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