DATE: Tuesday, April 13 (2010), at 4:05pm
PLACE: LIT 305
SPEAKER:
Krishnaswami Alladi
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TITLE:
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Some new observations on the Gollnitz-Gordon and Rogers-Ramanujan identities - II
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ABSTRACT:
In the previous lecture under the same title, we
showed that the odd and even parts of the Rogers-Ramanujan
functions possessed product representations mod 80. Then
using the Quintuple Product Identity, we showed how these
products are connected to certain series representations of
Andrews. Finally by exploiting the Quintuple Product Identity,
we deduced certain surprising shifted partition identities mod 30
from the Andrews series and our product representations.
My guru Basil Gordon has formulated a META THEOREM which
states that "What works for 5 works for 8". The Gollnitz-Gordon
identities are to the modulus 8 what the Rogers-Ramanujan
identities are to the modulus 5. Guided by this philosophy of
Gordon, we will use the Quintuple Product Identity to get new
proofs of the Gollnitz-Gordon identities from the odd-even split
of Euler's celebrated Pentagonal Numbers Theorem. We will then
exploit this approach to establish some unexpected shifted
partition identities mod 48, namely identities connecting the
number of partitions of n in certain residue classes mod 48 to
partitions of n-1 or n-2 in certain other residue classes mod 48.
EVEN THOUGH THIS TALK IS RELATED TO, AND IS A CONTINUATION
OF THE TALK GIVEN LAST WEEK, THIS TALK WILL BE SELF CONTAINED.
This talk will focus on other certain other parts of my paper
K. Alladi, "On some new observations on the Gollnitz-Gordon
and Rogers-Ramanujan identities", Trans. AMS 347
(1995), 897-914
not covered in the lecture last week.
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