Baruah--Gogoi关于奇数和偶数上划线部分之和的猜想的一个证明
A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts
- Shanghai University of International Business and Economics（上海对外经贸大学）
- Suzhou University of Science and Technology（苏州科技大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文用初等q-级数方法（结合Watson五重积恒等式特化、除数级数参数化与逐项对数微分）证明了Baruah-Gogoi关于超分拆奇偶上划线部分和统计量模7同余的猜想。
中文摘要 AI 辅助
Andrews和Dastidar引入了统计量$\SOME(n)$，定义为$n$的分拆中出现的所有奇数部分之和与所有偶数部分之和的差，并研究了其算术性质。受他们工作的启发，Baruah和Gogoi为超分拆定义了两种类似的统计量，即$\OSOMEo(n)$和$\OSOMEe(n)$，分别记录所有奇数和所有偶数上划线部分之和。他们建立了这些统计量及其差值的若干同余式，并提出了一个关于模$7$同余式的猜想。在本文中，我们通过初等的$q$-级数方法证明了他们的猜想。我们的证明结合了Watson五重积恒等式的两个特化、除数求和级数的$(p,k)$-参数化以及逐项对数微分。
英文摘要
Andrews and Dastidar introduced the statistic $\SOME(n)$, defined as the difference between the sum of all odd parts and the sum of all even parts occurring in the partitions of $n$, and investigated its arithmetic properties. Motivated by their work, Baruah and Gogoi defined two analogous statistics for overpartitions, namely $\OSOMEo(n)$ and $\OSOMEe(n)$, which record the sums of all odd and all even overlined parts, respectively. They established several congruences for these statistics and their difference, and proposed a conjecture on congruences modulo $7$. In this paper, we prove their conjecture by elementary $q$-series methods. Our proof combines two specializations of Watson's quintuple product identity, a $(p,k)$-parameterization of divisor-sum series, and termwise logarithmic differentiation
