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

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 >= 3 and n >= 0, we prove that c phi(3)(3(2k)n + 7 . 3(2k) + 1/8) 0 (mod 3(4k+5)). We give two different proofs to the congruences satisfied by c phi(9)(n). One of the proofs uses a relation between c phi(9)(n) and c phi(3)(n) due to Kolitsch, for which we provide a new proof in this paper. (C) 2018 Elsevier B.V. All rights reserved.