Congruences Modulo Powers of 3 for 3- and 9-Colored Generalized Frobenius Partitions

TL;DR

This paper establishes infinite families of congruences modulo powers of 3 for the counts of 3- and 9-colored generalized Frobenius partitions, refining previous results and providing new proofs for key relations.

Contribution

It introduces new infinite families of congruences for $c\phi_{3}(n)$ and $c\phi_{9}(n)$ modulo powers of 3, with novel proofs and refinements of earlier work.

Findings

Proves congruences for $c\phi_{3}(n)$ modulo $3^{4k+5}$.

Provides two proofs for congruences of $c\phi_{9}(n)$, including a new proof of a relation with $c\phi_{3}(n)$.

Refines previous results by Kolitsch on generalized Frobenius partitions.

Abstract

Let $c\phi_{k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c\phi_{3}(n)$ and $c\phi_{9}(n)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that \[c\phi_{3}\Big(3^{2k}n+\frac{7\cdot 3^{2k}+1}{8}\Big) \equiv 0 \pmod{3^{4k+5}}.\] We give two different proofs to the congruences satisfied by $c\phi_{9}(n)$. One of the proofs uses an relation between $c\phi_{9}(n)$ and $c\phi_{3}(n)$ due to Kolitsch, for which we provide a new proof in this paper.