*Abstract:* |
In his 1984 AMS Memoir, George Andrews defined two families of generalized Frobenius partition
functions which he denoted $\phi_k(n)$ and $c\phi_k(n)$ where $k\geq 1.$
Both of these functions "naturally" generalize the unrestricted partition function
$p(n)$ since $p(n) = \phi_1(n) = c\phi_1(n)$ for all $n.$
In his 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 generalized Frobenius partition functions, typically for small values of $k.$
In this talk, I will discuss a variety of these past congruence results
(including the recent work of Paule and Radu). I will then transition to very recent
work of Baruah and Sarmah who, in 2011, proved a number of congruence properties
for $c\phi_4$, all with moduli which are powers of 4.
I will then provide an elementary proof of a new congruence for $c\phi_4$ by
proving this function satisfies an unexpected result modulo 5.
(The proof relies on Baruah and Sarmah's results as well as work of Srinivasa Ramanujan.)
I will then close with comments about current and future work. |