Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

Abstract

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions of $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families of congruences modulo arbitrary powers of $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $n$ and $k$, we have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\mathrm{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}\cdot n+\dfrac{3^{2k+4}-13}{4}\right)&\equiv0~\left(\mathrm{mod}~3^{2k+5}\right).\end{align*}