Nayandeep Deka Baruah ; Hirakjyoti Das - Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

hrj:8927 - Hardy-Ramanujan Journal, January 9, 2022, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021 - https://doi.org/10.46298/hrj.2022.8927
Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

Authors: Nayandeep Deka Baruah ; Hirakjyoti Das

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*}

Volume: Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021
Published on: January 9, 2022
Accepted on: January 9, 2022
Submitted on: January 6, 2022
Keywords: l-regular partitions,Congruences,Generating functions,2010 Mathematics Subject Classification. 05A17; 11P83,[MATH]Mathematics [math]