DATE: Tuesday, April 6 (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 - I
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ABSTRACT:
The celebrated Rogers-Ramanujan identities
provide product representations mod 5 for a certain pair of
q-series. The ratio of these two q-series yields Ramanujan's
famous continued fraction. We show that the odd and even
parts of the Rogers-Ramanujan functions have product
representations mod 80. For this we utilize certain identities
of L. J. Rogers and a modular relation connecting them due to
Andrews. Previously. Andrews had obtained representations
for the odd and even parts of the Rogers-Ramanujan functions
in the form of series. We show that by combining the Andrews
series representations along with the Quintuple Product Identity
yields our product representations modulo 80. As a by-product
we show that this leads to some interesting shifted partition
identities connecting partitions of n in certain residue classes
modulo 30 to partitions of n-1 or n-3 in certain other residue
classes modulo 30. This talk will focus on a part of my paper
K. Alladi, "On some new observations on the Gollnitz-Gordon
and Rogers-Ramanujan identities", Trans. AMS 347
(1995), 897-914
In the second talk under the same title on Apr 13, I will discuss
related results arising from a study of the Gollnitz-Gordon identities.
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