AI 中文总结
本文研究仿模形式的卷积恒等式，推广了全纯投影理论的结果，解决了相关猜想并得到18个仿θ函数卷积恒等式，可简化为5个核心恒等式。
AI 中文摘要
卷积恒等式和递推公式在全纯模形式理论及其应用中早已发挥重要作用，它们既是引人注目的公式，也是组合学应用的基础有用工具。近来，人们对非全纯模形式产生的例子重新产生兴趣，包括可追溯至1885年的著名Hurwitz-Kronecker类数关系，以及Imamoğlu、Raum和Richter在2014年的开创性工作。Imamoğlu等人建立了（向量值）调和Maass形式与全纯模形式乘积的全纯投影理论，以生成大量此类公式，该工作随后被Duncan、Griffin和Ono与Conway和Norton意义下的可复制型函数联系起来，这类递推在他们对Umbral moonshine猜想的证明中发挥了关键作用。本文中，我们针对此类函数的全纯投影取得了新结果，这些结果对许多自然情形更为便利。Imamoğlu等人的选择对应广义Pell方程$m^2 - Dn^2 = N$中$D$取平方值的情形，对于给定的$N$，该情形仅有有限多解；我们的主要结果覆盖一般$D$的情形，尤其是Pell方程有无穷多解的情形。作为应用，我们解决了第二作者近期提出的一个猜想。更一般地，我们的方法得到了18个仿θ函数的卷积恒等式，可将其简化为5个恒等式，用于直接推导其余恒等式。在附录中，我们还展示了这些恒等式如何通过更直接的$q$-级数方法证明；不过，全纯投影公式的关键效用在于，它们提供了一种自动发现和验证此类公式的工具。
英文摘要
Convolution identities and recursive formulas have long played a role in the theory of holomorphic modular forms and their applications. These have served both as striking formulas and as fundamentally useful tools for applications to combinatorics. Recently, there has been renewed interest in examples arising from non-holomorphic modular forms. These include the famous Hurwitz-Kronecker class number relations dating to 1885, and groundbreaking work of Imamoğlu, Raum, and Richter from 2014. Imamoğlu, Raum, and Richter developed a theory of holomorphic projection for products of (vector-valued) harmonic Maass forms and holomorphic modular forms to produce many such formulas. This was related shortly thereafter by Duncan, Griffin, and Ono to replicable-type functions in the sense of Conway and Norton, and such recursions played a key role in their proof of the Umbral Moonshine Conjecture. Here, we develop new results on holomorphic projections of such functions which is more convenient for many natural cases. Imamoğlu, Raum, and Richter's choice corresponds to the case in which the generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, which has only finitely many solutions for a given value of $N$. Our main results cover the cases of general $D$, in particular those for which the Pell equation has infinitely many solutions. As an application, we resolve a recent conjecture of the second author. More generally, our approach yields 18 convolution identities for mock theta functions, which we boil down to 5 identities that can directly be used to prove the others. In the appendices, we also show how these identities can be proven by more direct $q$-series methods; however, the key utility of the holomorphic projection formulas is that they give a tool to automatically discover and verify such formulas.
Comments39 pages
