Congruence properties for a certain kind of partition functions

In light of the modular equations of fifth and seventh order, we derive some congruence properties for a certain kind of partition functions a(n) which satisfy Sigma(infinity)(n=0) a(n)q(n) equivalent to (q; q)(infinity)(k) (mod m), where k is a positive integer with 1 <= k <= 24 and m = 2,3. In view of these properties, we obtain many infinite families of congruences for c phi(k)(n), the number of generalized Frobenius partitions of n with k colors, and (c phi(k)) over bar (n), the number of generalized Frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors. Meanwhile, we also apply the main theorems to some other kinds of partition functions. (C) 2015 Elsevier Inc. All rights reserved.