Aequationes mathematicae

, Volume 81, Issue 1–2, pp 177–183 | Cite as

A q-rious positivity

  • S. Ole Warnaar
  • W. Zudilin


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.

Mathematics Subject Classification (2000)

Primary 11B65 Secondary 05A10 11B83 11C08 33D15 


Binomial coefficients q-binomial coefficients Gaussian polynomials factorial ratios basic hypergeometric series cyclotomic polynomials positivity 


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Authors and Affiliations

  1. 1.School of Mathematics and PhysicsThe University of QueenslandBrisbaneAustralia
  2. 2.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

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