Rogers-Ramanujan Type Identities

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Tuesday, March 14, 2023, 1 pm– 2 pm

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This is a past event.

Biography: Andrew Sills earned a B.A. from Rutgers, M.A. from Penn State, and Ph.D. from University of Kentucky, all in mathematics. His thesis advisor was George Andrews of Penn State. After a postdoctoral position at Rutgers under the mentorship of Doron Zeilberger, he began a tenure-track position at Georgia Southern University, attaining full professorship in 2015. He is the past recipient of an NSA research grant. His research interests include the theory of integers partitions, q-series, and related topics. Most recently, his research activities have expanded to include work in mathematical statistics and applications of mathematics to music.


Abstract: In 1894, L. J. Rogers published a paper on the expansion of certain infinite products that
included a pair of q-series identities that were later rediscovered independently by S. Ramanujan (at
first without proof). These analytic identities, traditionally written in the variable “q”, each asserted
the equality of an infinite product with an infinite series where the general term involved a power of
q and rising q-factorials. Ramanujan’s mentor G. H. Hardy was unaware of Rogers’ work, and
circulated the identities among various European mathematicians as unproven conjectures. Later
Ramanujan discovered a proof, and then came across Rogers’ earlier work in an old issue of the
Proceedings of the London Mathematical Society. These identities came to be known as the Rogers--
Ramanujan identities. In hindsight, some earlier identities due to Euler, Jacobi, and Heine could be
classified as identities “of Rogers--Ramanujan type.” Rogers and Ramanujan each discovered quite a
few identities of similar type in the 1910s, but then the subject remained relatively dormant until W.
N. Bailey (who knew Ramanujan while undergraduate at Cambridge) and his then-Ph.D. student L.
J. Slater revived the study of Rogers--Ramanujan type identities in the 1940s. These identities of
analytic functions can also be viewed as generating functions for classes of combinatorial objects
(often integer partitions). This combinatorial view became particularly popular in the 1960s under
the leadership of Basil Gordon and George Andrews, when a torrent of infinite families of
combinatorial identities of Rogers--Ramanujan type were discovered. Since the 1980s, many
additional applications of Rogers--Ramanujan type identities have been found in a variety of areas of
mathematics and physics. As time allows, I will discuss some recent advances and open problems,
including some of my own work in the area.

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