Old and New Results for Generalized Frobenius Partition Functions

Speaker: James Sellers
Date: 06/03/2013
Time: 14:00 - 16:00

Location: RISC Seminar room

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.