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

In his 1984 AMS Memoir, George Andrews defined the family of k–colored generalized Frobenius partition functions. These are enumerated by cφk(n) where k ≥ 1 is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all n ≥ 0, cφ2(5n+3) ≡ 0 (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 cφk(n) continues, with recent works completed by Baruah and Sarmah as well as the second 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 cφk(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 cφk(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.