On 2-color partitions where one of the color is multiples of $5^k$

Abstract

In this work, we investigate the arithmetic properties of $p_{1,5^k}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are multiples of $5^k$. By constructing generating functions for $p_{1,5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type infinite family of congruences modulo powers of $5$.