论文标题
Omega SPT一致性家族的单个可兑现的证明5
A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5
论文作者
论文摘要
在2018年，Liuquan Wang和Yifan Yang证明了与第三阶模拟Theta函数$ω（Q）$相对应的最小零件功能的无限家族。他们的证明采用了需要20个初始关系的入门形式，并利用模块化函数的空间同构为免费等级2 $ \ mathbb {z} {x] $ - $ - 模块。该证明策略最初是由Paule和Radu开发的，用于研究与属1的模块化曲线有关的一致性家族。我们表明，王和杨的一致性与属0模块化曲线相关的一致性可以通过单个可行的方法证明，这是通过单向函数通过模块化函数的$ \ nathbbbbb的本地化来证明的。据我们所知，这是第一次将这种代数结构应用于分区一致性理论。我们的诱导更加复杂，并依赖于表现出一些不规则5种生长的功能序列。但是，证明最终仅取决于仅对10个初始关系的直接验证，并且类似于Ramanujan和Watson的经典方法。
In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $ω(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.
