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A \(q\)-rious positivity. (English) Zbl 1234.11023

The \(q\)-factorial of an integer \(n\geq 1\) is defined by \[ [n]!=\prod_{i=1}^n\frac{1-q^i}{1-q}. \] For \({\mathbf a}=(a_1,\dots,a_r)\) and \({\mathbf b}=(b_1,\dots,b_s)\) tuples of positive integers satisfying \[ \sum_{i=1}^r\lfloor a_i x \rfloor-\sum_{j=1}^s\lfloor b_j x \rfloor\geq 0,\forall x\geq 0 \] the authors conjecture that the polynomial \[ D({\mathbf a},{\mathbf b};q):=\frac{[a_1]!\dots[a_r]! }{[b_1]!\dots[b_s]!} \] is positive. The conjecture is true for the \(q\)-super Catalan numbers \[ A_{n,m}(q)=D((2n,2m),(n,n+m,m);q),n,m\geq 0 \] and for \[ B_{n,m}(q):=D((2n,m),(n,2m,n-m);q),n\geq m\geq 0. \]

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
11B83 Special sequences and polynomials
11C08 Polynomials in number theory
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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