Abstract We extend some results recently obtained by Dan Romik [14] about the Taylor coefficients of the theta function θ 3 ( e − π ) to the case θ 3 ( q ) of a real valued variable 0 q 1 . These results are obtained by carefully studying the properties of the cumulants associated to a Jacobi θ 3 (or discrete normal) distributed random variable. This article also states some integrality conjectures about rational sequences that generalize the one studied by Romik.

中文翻译：

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Jacobi θ3(q) 函数的泰勒系数

摘要 我们将 Dan Romik [14] 最近获得的关于 theta 函数 θ 3 ( e − π ) 的泰勒系数的一些结果扩展到实值变量 0 q 1 的情况 θ 3 ( q )。这些结果是通过仔细研究与 Jacobi θ 3（或离散正态）分布随机变量相关联的累积量的特性而获得的。这篇文章还陈述了一些关于有理序列的完整性猜想，这些猜想概括了 Romik 所研究的序列。