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

TL;DR

This paper analyzes the 2-color partition function $p_{1,5^k}(n)$, counting partitions where one color uses only parts divisible by $5^k$, and derives Ramanujan-type congruences modulo powers of $5$ by constructing specialized generating functions. Leveraging a Huffing operator modulo $5$ and 5-adic valuation techniques, the authors propagate recursive coefficient structures to obtain infinite families of congruences indexed by $k$ and an auxiliary parameter $eta$. The main contributions are explicit congruence families for all $k\ge1$ and $\beta\ge0$, generalizing and unifying earlier results (e.g., for $k=1,2$) and extending Ramanujan-type phenomena to broader arithmetic progressions. The work advances partition theory and modular-form-like congruences by providing a cohesive framework that links generating-function decompositions, 5-adic valuations, and infinite congruence families.

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 $5$.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1: HH
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1: HH, Lemma 4.1
• Lemma 4.2: HH, Lemma 4.3