Generalized Frobenius Partitions Modulo Powers of $2$

RESUMO

Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $c\phi_{18}(2n+1)$ modulo $16$ completely. In particular, we also prove \[ \sum_{n\geq0}c\phi_{18}(6n+1)q^n \equiv4\sum_{r\in\mathbb{Z}}q^{r(3r-1)/2}\pmod{16}, \] which proves the congruences $c\phi_{18}(30n+19)\equiv c\phi_{18}(30n+25)\equiv0\pmod{16}$ 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 $c\phi_{10}(2n+1)$ modulo $8$.