Congruences for Generalized Frobenius Partitions with an 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 $c{\varphi }_k(n)$ where $k\ge 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\ge 0,$ $c{\varphi }_2(5n+3)\equiv 0(\mathrm{mod}5).$ 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 $c{\varphi }_k(n)$ 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 $c{\varphi }_k(n)$ 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 $c{\varphi }_k(n)$ 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.

References in corpus (2)

• An Unexpected Congruence Modulo 5 for 4--Colored Generalized Frobenius Partitions
• Congruences modulo powers of 5 for three-colored Frobenius partitions