Congruences modulo powers of 5 for $k$-colored partitions

TL;DR

This paper establishes infinite families of congruences modulo powers of 5 for the number of k-colored partitions, extending Ramanujan-type results to new cases with explicit formulas and proofs.

Contribution

It introduces new infinite families of congruences modulo powers of 5 for k-colored partitions, including explicit formulas for specific values of k.

Findings

Proves congruences modulo 25 for k-colored partitions.

Establishes Ramanujan-type congruences for k=2, 6, 7.

Provides explicit formulas for congruences involving powers of 5.

Abstract

Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for $p_{-k}(n)$ with $k=2, 6$, and $7$. For example, for all integers $n\geq0$ and $\alpha\geq1$, we prove that \begin{align*} p_{-2}\left(5^{2\alpha-1}n+\dfrac{7\times5^{2\alpha-1}+1}{12}\right) &\equiv0\pmod{5^{\alpha}} \end{align*} and \begin{align*} p_{-2}\left(5^{2\alpha}n+\dfrac{11\times5^{2\alpha}+1}{12}\right) &\equiv0\pmod{5^{\alpha+1}}. \end{align*}