Elementary Proofs of Congruences for the Cubic and Overcubic Partition Functions

Combinatorics/Partitions Seminar

Meeting Details

For more information about this meeting, contact James Sellers, Matthew Katz, George Andrews.

Speaker: James Sellers, PSU

Abstract: In 2010, Hei-Chi Chan introduced the cubic partition function $a(n)$ in connection with Ramanujan's cubic continued fraction. Among other things, Chan proved that, for all $n\geq 0,$ $a(3n+2) \equiv 0 \pmod{3}.$ In the same year, Byungchan Kim introduced the overcubic partition function $\overline{a}(n).$ Using modular forms, Kim found a generating function representation for \overline{a}(3n+2) which implies that $\overline{a}(3n+2) \equiv 0 \pmod{6}$ for all $n\geq 0.$ More recently, Hirschhorn has proven Kim's generating function result using elementary generating function manipulations. In this talk, we use elementary means to prove functional equations satisfied by the generating functions for $a(n)$ and $\overline{a}(n),$ respectively. These lead to new representations of these generating functions as products of terms involving Ramanujan's $\psi$ and $\varphi$ functions. In the process, we are able to prove the congruences mentioned above as well as numerous arithmetic properties satisfied by $\overline{a}(n)$ modulo small powers of 2 which do not appear in the literature.

Room Reservation Information

Room Number: 106 McAllister

Date: 12/10/2013

Time: 10:10am - 11:00am