![[Abstract]](http://qseries.org/fgarvan/icons/scroll.gif){kind=icon}
F.G. Garvan and J.A. Sellers;
Congruences for generalized Frobenius partitions with an arbitrarily
large number of colors
,
INTEGERS, 14 (2014), Article A7.
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.
The url of this page is http://qseries.org/fgarvan/abstracts/gen-frob-congs.html.
fgarvan@ufl.edu
|