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Some congruences for 12-coloured generalized Frobenius partitions
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 3 / August 2024
- Published online by Cambridge University Press:
- 02 May 2024, pp. 778-793
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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 
$c\phi_{12}(n)$.
