Generalized Frobenius Partitions Modulo Powers of $2$

Abstract

Let $c{\varphi }_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\ge 2$ and every $k\equiv 2(\mathrm{mod}{2}^m)$, \[ \sum_{n\geq0}cϕ_k(n)q^n\equiv\frac{φ(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}cϕ_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\phi (q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $c{\varphi }_{18}(2n+1)$ 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 $c{\varphi }_{18}(30n+19)\equiv c{\varphi }_{18}(30n+25)\equiv 0(\mathrm{mod}16)$ 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 $c{\varphi }_{10}(2n+1)$ modulo $8$.

arXiv comment12 pages, comments are welcome