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Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2020-07-21 , DOI: 10.1142/s1793042120400126 Amanda Folsom 1
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2020-07-21 , DOI: 10.1142/s1793042120400126 Amanda Folsom 1
Affiliation
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:
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。
更新日期:2020-07-21
中文翻译:
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