Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors

TL;DR

This paper establishes new infinite families of congruences for generalized Frobenius partition functions with arbitrarily many colors, extending Ramanujan-like congruences to large values of k using elementary methods.

Contribution

It proves the first Ramanujan-like congruences for generalized Frobenius partitions with large k, using a generating function approach that was previously underutilized.

Findings

Proves infinite families of congruences for cφ_k(n) with large k.

Uses elementary generating function techniques from Andrews' Memoir.

Extends Ramanujan-like congruences to arbitrarily large number of colors.

Abstract

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $c\phi_2(5n+3) \equiv 0\pmod{5}.$ Soon after, many authors proved congruence properties for various $k$--colored generalized Frobenius partition functions, typically with a small number of colors. Work on Ramanujan--like congruence properties satisfied by the functions $c\phi_k(n)$ continues, with recent works completed by Baruah and Sarmah as well as the author. Unfortunately, in all cases, the authors restrict their attention to small values of $k.$ This is often due to the difficulty in finding a "nice" representation of the generating function for $c\phi_k(n)$ for large $k.$ Because of this, no Ramanujan--like congruences are known where $k$ is large. In this note, we rectify this situation by proving several infinite families of congruences for $c\phi_k(n)$ where $k$ is allowed to grow arbitrarily large. The proof is truly elementary, relying on a generating function representation which appears in Andrews' Memoir but has gone relatively unnoticed.