Abstract
The q-binomial coefficients \({\genfrac{[}{]}{0pt}{}{n}{m}= \prod_{i=1}^m (1-q^{n-m+i})/(1-q^i)}\), for integers 0 ≤ m ≤ 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 \({\genfrac{[}{]}{0pt}{}{n}{m}}\). 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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Warnaar, S.O., Zudilin, W. A q-rious positivity. Aequat. Math. 81, 177–183 (2011). https://doi.org/10.1007/s00010-010-0055-9
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DOI: https://doi.org/10.1007/s00010-010-0055-9