Abstract:

The so-called "First Borwein Conjecture" concerns a very easy-to-state assertion in the theory of $q$-series that was originally conjectured by Peter Borwein in 1990 and first published by George Andrews in 1995. Since then, despite having received a lot of attention by various experts in $q$-series, the conjecture has resisted many attempts to prove it. Only recently (in a 2019 preprint), Chen Wang, a student of Christian Krattenthaler at the University of Vienna, successfully vanquished the First Borwein Conjecture using analytic methods. In my talk I will present an infinite family of Borwein type conjectures (related to partial theta products) that go beyond the now classical First Borwein Conjecture. In order to get a better feeling for the new conjectures, I shall give various background information and discuss some of the surrounding theory: $q$-series, Jacobi's triple product identity, Jacobi theta functions, Macdonald polynomials, multiple $q$-series, etc. This talk is based on recent joint work with Gaurav Bhatnagar: arxiv.org/abs/1902.04447