Some congruences for 12-coloured generalized Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the family of functions cφₖ(n) , the number of k -coloured generalized Frobenius partitions of n . In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of CΦₖ(q) for 2≤ k≤17 by utilizing the theory of modular forms, where CΦₖ(q) denotes the generating function of cφₖ(n) . In this paper, we first establish another expression of CΦ₁₂(q) with integer coefficients, then prove some congruences modulo small powers of 3 for cφ₁₂(n) by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by cφ₁₂(n) .