A relationship between certain colored generalized Frobenius partitions and ordinary partitions

Abstract

Ramanujan's congruence p(5 n + 4) ≡ 0 (mod 5) for ordinary partitions is well-known. This congruence is just the first in a family of congruences modulo 5; namely, p(5 n n + δ α ) ≡ 0 (mod 5 α ) for α ≧ 1 where δ α represents the reciprocal of 24 modulo 5 α. A similar family of congruences exists for ordinary partitions modulo 7. In this paper we prove the corresponding congruences for generalized Frobenius partitions with 5 and 7 colors modulo 5 and 7, respectively, by establishing an equality between these two classes of generalized Frobenius partitions and certain ordinary partitions. The proofs are based on some elegant identities of Ramanujan.