Let $c\phi_{ k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c\phi_{ 3}(n)$ and $c\phi_{ 9}(n)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that \[c\phi_{ 3}\Big(3^{ 2k}n+\frac{ 7\cdot 3^{ 2k}+1}{ 8}\Big) \equiv 0 \pmod{ 3^{ 4k+5}}.\] We give two different proofs to the congruences satisfied by $c\phi_{ 9}(n)$. One of the proofs uses an relation between $c\phi_{ 9}(n)$ and $c\phi_{ 3}(n)$ due to Kolitsch, for which we provide a new proof in this paper.
Categories: math.CO math.NT