奇部超额分拆元组的模$2$幂内部同余
Internal congruences modulo powers of $2$ for overpartition tuples with odd parts
- Ahmedabad University（艾哈迈达巴德大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明奇部超额分拆元组数在$2$幂指标下的内部同余，通过统一的三项递推多项式族给出初等证明，并精确刻画$2$-进赋值等于奇平方数。
AI中文摘要：
设$\overline{\mathrm{OPT}}_m(n)$表示$n$的奇部超额分拆$m$-元组数。我们证明：对每个奇数$m\ge1$和每个$i\ge3$，有\\[\sum_{n\ge0}\Bigl(\overline{\mathrm{OPT}}_m\bigl(2^in\bigr)-\overline{\mathrm{OPT}}_m\bigl(2^{i-1}n\bigr)\Bigr)q^n \equiv 2^{\\,i+1}\sum_{k\ge0}q^{(2k+1)^2} \pmod{2^{\\,i+2}}.\\]因此$\overline{\mathrm{OPT}}_m(2^in)\equiv \overline{\mathrm{OPT}}_m(2^{i-1}n)\pmod{2^{i+1}}$，且$2$-进赋值恰好等于奇平方数时取等号。证明是初等的且对$m$一致：由三项递推给出的单个整数多项式族控制所有涉及的$U$-算子恒等式，一个整除性陈述在每次迭代中提供一个$2$的幂。
英文摘要：
Let $\overline{\mathrm{OPT}}_m(n)$ denote the number of overpartition $m$-tuples of $n$ into odd parts. We prove that for every odd $m\ge1$ and every $i\ge3$, \[\sum_{n\ge0}\Bigl(\overline{\mathrm{OPT}}_m\bigl(2^in\bigr)-\overline{\mathrm{OPT}}_m\bigl(2^{i-1}n\bigr)\Bigr)q^n \equiv 2^{\,i+1}\sum_{k\ge0}q^{(2k+1)^2} \pmod{2^{\,i+2}} .\] Thus $\overline{\mathrm{OPT}}_m(2^in)\equiv \overline{\mathrm{OPT}}_m(2^{i-1}n)\pmod{2^{i+1}}$, with equality of $2$-adic valuations exactly at the odd squares. The proof is elementary and uniform in $m$: a single family of integer polynomials, given by a three-term recurrence, governs every $U$-operator identity involved, and a divisibility statement supplies one power of $2$ per iteration.
