A q-rious positivity

Details A q-rious positivity A q-rious positivity

2010-03-09

arxiv.org/abs/1003.1999v1

Aequat. Math. 81 (2011), no. 1-2, 177--183

Mathematics

Number Theory

6 pages

Scientific paper

10.1007/s00010-010-0055-9

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, 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 $\qbinom{n}{m}$. In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of $q$-factorials that happen to be polynomials.

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