负变元高斯二项式系数
Gaussian binomial coefficients with negative arguments
摘要：
勒布表明，通常的二项式系数自然扩张到负（整数）条目继续满足许多基本性质。特别是，他给了一个统一的二项式定理以及组合的解释方面选择子集的集合与负数的元素。我们表明，所有这些都可以扩展到高斯二项式系数的情况下。此外，我们证明了几个著名的算术性质的二项式系数也持有的情况下，负的条目。特别是，我们表明，卢卡斯定理的二项式系数模$p$不仅自然延伸到负条目的情况下，但甚至高斯的情况下。
英文摘要：
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.
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