Rogers-Ramanujan Type Identities

Project Number
RI 3/12 TPC

Project Duration
April 2013 - April 2017

Status
In-Progress (Extended)

Abstract
The theory of q-series lies at the intersection of three major areas in mathematics: combinatorics, number theory and special functions. Perhaps the most representative examples of q-series are the Rogers--Ramanujan identities. Independently discovered by Rogers (1894) and Ramanujan (1919), the Rogers--Ramanujan identities have since been studied and generalized in different ways by many mathematicians including MacMahon (1916), Gordon (1961), Andrews (1974), Bressoud (1979,1980), Lepowsky with Milne (1978) and with Wilson (1982),), Warnaar with Schilling (1998) and with Zudilin (2012). Notably, Baxter (1980) had independently rediscovered these identities in the context of statistical mechanics, a branch of physics. Specifically, he found that Rogers--Ramanujan type identities arose as solutions to the Hard Hexagon Model and he received the Boltzman medal in 1980 for this important achievement. An account of the importance of these identities to physics can be found in the text of McCoy's invited lecture titled "Rogers--Ramanujan identities: a century of progress from mathematics to physics" at the 1998 International Congress of Mathematicians. The purpose of this project is to build upon existing knowledge to develop techniques for constructing new Rogers--Ramanujan type identities. We propose to do this through the study of q-trinomial coefficients. Andrews and Baxter (1987) had previously defined six possible analogues for q-trinomial coefficients. Warnaar (2003) followed up on their work by refining two of these q-trinomial coefficients. For these refinements, he produced a pair of iterative transformations that had far-reaching consequences. Warnaar's transformations were used to prove several new families of Rogers--Ramanujan type identities and also to achieve progress in what is known as the generalized Borwein conjecture. We have recently discovered another pair of transformations for the refined q-trinomial coefficients that were not considered by Warnaar. We believe that new families of Rogers--Ramanujan identities can be constructed with them and hope that the resulting work could shed new light on the original Borwein conjecture. As our study is still in the preliminary stages we are applying for funding to create opportunities for collaboration with international experts in the area. considered by Warnaar. We believe that new families of Rogers--Ramanujan identities can be constructed with them and hope that the resulting work could shed new light on the original Borwein conjecture. As our study is still in the preliminary stages we are applying for funding to create opportunities for collaboration with international experts in the area.

Funding Source
NIE

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