Euler's pentagonal numbers theorem - refinements, variations and companions

School of Mathematics Colloquium
Thursday, October 28, 2010 - 11:05
1 hour (actually 50 minutes)
Skiles 249
University of Florida
Euler's celebrated pentagonal numbers theorem is one themost  fundamental in the theory of partitions and q-hypergeometric series.The recurrence formula that it yields is what MacMahon used to compute atable of values of the partition function to verify the deep Hardy-Ramanujanformula. On seeing this table, Ramanujan wrote down his spectacular partition congruences. The author recently proved two new companions to Euler'stheorem in which the role of the pentagonal numbers is replaced by the squares.These companions are deeper in the sense that lacunarity can be achievedeven with the introduction of a  parameter.  One of these companions isdeduced from a partial theta identity in Ramanujan's Lost Notebook and theother from a q-hypergeometric identity of George Andrews. We will explainconnections between our companions and various classical results such asthe Jacobi triple product identity for theta functions and the partitiontheorems of Sylvester and Fine. The talk will be accessible to non-experts.