广义Frobenius分拆模$2$的幂
Generalized Frobenius Partitions Modulo Powers of $2$
- 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：
本文证明广义Frobenius分拆数模$2$的幂的同余式，完全确定$c\phi_{18}(2n+1)$模$16$和$c\phi_{10}(2n+1)$模$8$的值，并证明Das等人猜想。
AI中文摘要：
设$c\phi_k(n)$表示$n$的$k$色广义Frobenius分拆数。我们证明，对于每个$m\geq2$和每个$k\equiv2\pmod{2^m}$，有\\[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \\]其中$\varphi(q)$是经典theta函数。对于$m=2$，这恢复了Chan、Wang和Yang的一个同余式。应用于$k=18$时，我们完全确定了$c\phi_{18}(2n+1)$模$16$的值。特别地，我们还证明了\\[ \sum_{n\geq0}c\phi_{18}(6n+1)q^n \equiv4\sum_{r\in\mathbb{Z}}q^{r(3r-1)/2}\pmod{16}, \\]这证明了Das、Nath和Sarma（2026）最近 conjectured 的同余式$c\phi_{18}(30n+19)\equiv c\phi_{18}(30n+25)\equiv0\pmod{16}$。它还产生了模$16$的进一步同余式和一个简单的模$8$刻画，该刻画恢复并扩展了这些作者的一个近期同余式。作为第二个应用，我们取$k=10$并确定$c\phi_{10}(2n+1)$模$8$的值。
英文摘要：
Let $cϕ_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}cϕ_k(n)q^n\equiv\frac{φ(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}cϕ_{k/2}(n)q^{2n}\pmod{2^m}, \] where $φ(q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $cϕ_{18}(2n+1)$ modulo $16$ completely. In particular, we also prove \[ \sum_{n\geq0}cϕ_{18}(6n+1)q^n \equiv4\sum_{r\in\mathbb{Z}}q^{r(3r-1)/2}\pmod{16}, \] which proves the congruences $cϕ_{18}(30n+19)\equiv cϕ_{18}(30n+25)\equiv0\pmod{16}$ recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set $k=10$ and determine $cϕ_{10}(2n+1)$ modulo $8$.
