Abstract
Abstract
In his 1984 AMS Memoir, Andrews introduced the family of functions
$c\phi_k(n)$
, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of
$\textrm{C}\Phi_k(q)$
for
$2\leq k\leq17$
by utilizing the theory of modular forms, where
$\textrm{C}\Phi_k(q)$
denotes the generating function of
$c\phi_k(n)$
. In this paper, we first establish another expression of
$\textrm{C}\Phi_{12}(q)$
with integer coefficients, then prove some congruences modulo small powers of 3 for
$c\phi_{12}(n)$
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
$c\phi_{12}(n)$
.
Publisher
Cambridge University Press (CUP)