ωSPT同余族在5次幂上的单变量证明。

IF 0.6 3区 数学 Q3 MATHEMATICS
Ramanujan Journal Pub Date : 2023-01-01 Epub Date: 2023-06-28 DOI:10.1007/s11139-023-00747-9
Nicolas Allen Smoot
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引用次数: 5

摘要

2018年,王柳泉和杨一凡证明了三阶模拟θ函数ω(q)对应的最小部分函数存在一个无穷同余族。他们的证明采用了需要20个初始关系的归纳的形式,并利用了同构于自由秩2Z[X]模的模函数空间。这种证明策略最初是由Paule和Radu开发的,用于研究与亏格1的模曲线相关的同余族。我们证明了与亏格0模曲线相关的王和杨同余族,可以通过同构于Z[X]的局部化的模函数环,使用单变量方法来证明。据我们所知,这是第一次将这样的代数结构应用于配分同余理论。我们的归纳更为复杂,并且依赖于表现出某种不规则的五元收敛的函数序列。然而,该证明最终仅基于对10个初始关系的直接验证,并且类似于Ramanujan和Watson的经典方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A single-variable proof of the omega SPT congruence family over powers of 5.

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 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 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 convergence. 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.

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来源期刊
Ramanujan Journal
Ramanujan Journal 数学-数学
CiteScore
1.40
自引率
14.30%
发文量
133
审稿时长
6-12 weeks
期刊介绍: The Ramanujan Journal publishes original papers of the highest quality in all areas of mathematics influenced by Srinivasa Ramanujan. His remarkable discoveries have made a great impact on several branches of mathematics, revealing deep and fundamental connections. The following prioritized listing of topics of interest to the journal is not intended to be exclusive but to demonstrate the editorial policy of attracting papers which represent a broad range of interest: Hyper-geometric and basic hyper-geometric series (q-series) * Partitions, compositions and combinatory analysis * Circle method and asymptotic formulae * Mock theta functions * Elliptic and theta functions * Modular forms and automorphic functions * Special functions and definite integrals * Continued fractions * Diophantine analysis including irrationality and transcendence * Number theory * Fourier analysis with applications to number theory * Connections between Lie algebras and q-series.
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