Legendre Theorems, Mock theta functions and Overpartitions

When/Where:

January 19, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

This is a report on two papers which are joint work with Atul Dixit and Ae Ja Yee.  We begin by studying the generating function for partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part. Surprisingly, the generating function turns out to be \(\omega(q)q\) (resp. \(\nu(-q)\), where \(\omega(q)\) and \(\nu(q)\) are two of the third order mock theta functions of Ramanujan.  This work is related to recent work of Garvan and Jennings-Shaffer.  There are neat Legendre theorems for both \(\omega(q)\) and \(\nu(q)\), and this leads to comparable Legendre theorems for subclasses of overpartitions that also naturally arise from the Garvan/Jennings-Shaffer work.