Ramanujan special talk: 10 years of q-rious positivity. More needed!

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.

classical analysis and ODEscombinatoricsnumber theory

Audience: advanced learners

Special Functions and Number Theory seminar

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