Krishnaswami Alladi (University of Florida)
"New observations on the Gollnitz-Gordon
and Rogers-Ramanujan identities"
ABSTRACT: In the entire theory of partitions and q-series,
the Rogers-Ramanujan identities are unmatched in simplicity,
elegance, and depth. The Gollnitz-Gordon identities are
to the modulus 8 what the Rogers-Ramanujan identities
are to the modulus 5. By considering the odd-even bisection
of certain fundamental theta function identities and by
evaluating these components in different ways, we will give
two new and quick proofs of the Gollnitz-Gordon identities.
We will then use such bisections to show that the odd and even
parts of the celebrated Rogers-Ramanujan identities and
the Rogers-Ramanujan continued fraction are products
modulo 80. Finally we will indicate how this approach
yields new shifted partition identities.