A q-rious positivity
arXiv:1003.1999 · doi:10.1007/s00010-010-0055-9
Abstract
The -binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers , are known to be polynomials with non-negative integer coefficients. This readily follows from the -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 -factorials that happen to be polynomials.
6 pages