DATE: Tuesday, August 31 (2010), at 1:55pm
PLACE: LIT 368
SPEAKER:
Krishna Alladi
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TITLE:
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A new companion to Euler's pentagonal numbers theorem
from partial theta identities of Andrews and Ramanujan
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ABSTRACT:
Euler's celebrated Pentagonal Numbers Theorem states that
if the set of partitions of an integer N into distinct parts is split into two
subsets based on the parity of the number of parts, then the split is even
except at the pentagonal numbers where the difference is 1. We will establish
a new companion to this result by considering partitions into distinct parts
with smallest part odd and show that when we consider a similar parity
split, the pentagonals are replaced by the squares! We deduce this as a
consequence of a partial theta identity in Ramanujan's Lost Notebook.
Actually Ramanujan's identity leads to a refinement with a free parameter
thereby making this companion stronger than Euler's theorem. We will
provide a novel proof of Ramanujan's partial theta identity and from the
method of the proof we will deduce a companion identity to Ramanujan's.
The talk will be accessible to non-experts.
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