Elementary proofs of congruences for the cubic and overcubic partition functions

In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan’s cubic continued fraction. Chan proved that ∑ n≥0 a(3n+ 2)q = 3 ∏ i≥1 (1− q3n)3(1− q) (1− qn)4(1− q2n)4 which clearly implies that, for all n ≥ 0, a(3n+ 2) ≡ 0 (mod 3). In the same year, Byungchan Kim introduced the overcubic partition function a(n). Using modular forms, Kim proved that ∑ n≥0 a(3n + 2)q = 6 ∏ i≥1 (1− q3n)6(1− q) (1− qn)8(1− q2n)3 . More recently, Hirschhorn has proven Kim’s generating function result above using elementary generating function methods. Clearly, this generating function result implies that a(3n+ 2) ≡ 0 (mod 6) for all n ≥ 0. In this note, we use elementary means to prove functional equations satisfied by the generating functions for a(n) and a(n), respectively. These lead to new representations of these generating functions as products of terms involving Ramanujan’s ψ and φ functions. In the process, we are able to prove the congruences mentioned above as well as numerous arithmetic properties satisfied by a(n) modulo small powers of 2.