On 2-color partitions with one color restricted to 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 congruences modulo powers of 5. The congruences previously derived by Wang [7] and Ranganath [6] can be seen as specific instances of the results presented in this work.