Arithmetic Properties for $k$-Color Analogue of Simultaneously $s$-Regular and $t$-Distinct Partitions

Abstract

In this article, we discuss general generating functions for partitions of $n$, simultaneously $s$-regular and $t$-distinct in 3-colors. In addition, we obtain infinite families of congruences modulo powers of 3 for specific values of $(\ell,t)$. For instance, for positive integers $n$ and $k$, we have \begin{align*} \sum_{n=o}^{\infty}RD_3^{3,3}\left(3^kn+\frac{3^k+1}{2}\right)q^n\equiv0 \pmod{3^{k+1}}. \end{align*}