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Two new families of q-positive integers

Sharon J. X. Hou·Jiang Zeng
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Abstract

Let $n,p,k$ be three positive integers. We prove that the rational fractions of $q$: $${n \brack k}_{q} {}_3ϕ_{2} [ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}| q;q^{k+1}]\quad\textrm{and}\quad q^{(n-p)p}\qbi{n}{k}{q} {}_3ϕ_2[ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}|q;q] $$ are polynomials of $q$ with positive integer coefficients. This generalizes a recent result of Lassalle (Ann. Comb. 6(2002), no. 3-4, 399-405), in the same way as the classical $q$-binomial coefficients refine the ordinary binomial coefficients.

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Sharon J. X. Hou, Jiang Zeng. 2004-03-05. Two new families of q-positive integers. https://arxiv.org/abs/math/0403064

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