A Survey of Classical Mock Theta Functions
A Survey of Classical Mock Theta Functions
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DOI：
10.1007/978-1-4614-0028-8_9
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
B. Gordon;Richard J. McIntosh
中科院分区：
文献类型：
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作者：
B. Gordon;Richard J. McIntosh
在他给哈代的最后一封信中，拉马努金定义了17个函数M（q），|Q| < 1，他称之为模拟θ函数。他观察到，作为qradially接近任何根的单位<$其中M（q）有一个指数奇点，有一个θ-功能T <$（q）与。从那时起，已经发现了具有这种性质的其他函数。我们列出了这些函数之间的各种线性关系，并在模群下给出了它们的变换律。我们证明了每一个mock θ-函数都与泛族（mock θ-代数）的一个成员有关。近年来，这一主题通过与马斯形式理论的紧密联系而获得了新的动力和重要性。本概览的最后一节简要介绍了这些新的发展情况。
In his last letter to Hardy, Ramanujan defined 17 functionsM(q), |q| < 1, which he called mock θ-functions. He observed that asqradially approaches any root of unity ζ at whichM(q) has an exponential singularity, there is a θ-functionTζ(q) with. Since then, other functions have been found which possess this property. We list various linear relations between these functions and develop their transformation laws under the modular group. We show that each mock θ-function is related to a member of a universal family (mock θ-conjectures). In recent years the subject has received new impetus and importance through a strong connection with the theory of Maass forms. The final section of this survey provides some brief remarks concerning these new developments.