Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas’ theorem on binomial coefficients modulo p not only extends naturally to the case of negative entries, but even to the Gaussian case.

中文翻译：

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带负参数的高斯二项式系数

勒布（Loeb）表明，将通常的二项式系数自然扩展到负（整数）项，仍然可以满足许多基本属性。特别是，他在选择具有负数个元素的集合的子集方面给出了统一的二项式定理以及组合解释。我们表明，所有这些都可以扩展到高斯二项式系数的情况。此外，我们证明了在负项的情况下，二项式系数的一些众所周知的算术特性也成立。特别地，我们证明了卢卡斯关于二项式系数模p的定理不仅自然地扩展到负项的情况，甚至扩展到高斯情况。