Old and New Results for Generalized Frobenius Partition Functions

Old and New Results for Generalized Frobenius Partition Functions

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.