Title:  Old and New Results for Generalized Frobenius Partition Functions

Speaker: Prof. James Sellers (PennS tate University)

Time and Location: Wednesday, March 6, 2013,
                   2:00 pm, RISC Seminarroom, Hagenberg

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.