A very happy new year to all. We have decided that the first talk
of every year will be a Ramanujan Special Talk. This year a colloquium
talk will be given by Wadim Zudilin. The announcement is below.
We
wish you many new theorems, ideas and papers in 2021. Please do send
any ideas or suggestions you have for the organisers to make this
seminar more successful and help serve the interests of this community.
Talk Announcement
Title: 10 years of q-rious positivity. More needed!
Speaker: Wadim Zudilin (Radboud University, Nijmegen).
Date and Time: Thursday, January 7, 2021, 3:55 PM IST (GMT+5:30)
Tea or coffee: Bring your own.
Where: Zoom: Please write to sfandnt@gmail.com for a link at-least 24 hours before the talk.
Abstract:
The $q$-binomial coefficients \[ \prod_{i=1}^m(1-q^{n-m+i})/(1-q^i),\] for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of them. Ten years ago, together with Ole Warnaar we observed that this non-negativity (aka positivity) property generalises to products of ratios of $q$-factorials that happen to be polynomials; we prove this observation for (very few) cases. During the last decade a resumed interest in study of generalised integer-valued factorial ratios, in connection with problems in analytic number theory and combinatorics, has brought to life new positive structures for their $q$-analogues. In my talk I will report on this "$q$-rious positivity" phenomenon, an ongoing project with Warnaar.
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