Some congruences for 12-coloured generalized Frobenius partitions

Abstract In his 1984 AMS Memoir, Andrews introduced the family of functions cₖ (n), the number of k -coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of Cₖ (q) for 2 k17 by utilizing the theory of modular forms, where Cₖ (q) denotes the generating function of cₖ (n). In this paper, we first establish another expression of C₁₂ (q) with integer coefficients, then prove some congruences modulo small powers of 3 for c₁₂ (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 c₁₂ (n).