In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as q tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function f(q) were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as q tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

中文翻译：

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模拟模和模形式的渐近展开、偏theta函数和径向极限差

1920 年，Ramanujan 研究了他的模拟 theta 函数和模 theta 函数之间的渐近差异，如q从单位圆盘内径向趋向于单位奇点的根。2013 年，Ramanujan 预测的关于他的模拟 theta 函数的有界渐近差异F(q)由 Ono、Rhoades 和作者建立，作为更一般结果的特例，其中它们被实现为量子模形式的特殊值。我们在这里的结果是三重的：我们将这些径向极限差异实现为偏 theta 函数的特殊值，为偏 theta 函数提供完全渐近展开为q趋向于径向单位根，并明确地将单位根处的偏 theta 函数评估为单位根的简单有限和。