Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions

Let cϕk(n) be the number of k-colored generalized Frobenius partitions of n. We establish some infinite families of congruences for cϕ3(n) and cϕ9(n) modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for k≥3 and n≥0, we prove that cϕ3(32kn+7⋅32k+18)≡0(mod34k+5).We give two different proofs to the congruences satisfied by cϕ9(n). One of the proofs uses a relation between cϕ9(n) and cϕ3(n) due to Kolitsch, for which we provide a new proof in this paper.