Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we bring together these two notions by introducing q-analogues of binomial coefficients of words. We study their basic properties, e.g., by extending classical formulas such as the q-Vandermonde and Manvel's et al. identities to our setting. As a consequence, we get information about the structure of the considered words: these q-deformations of binomial coefficients of words contain much richer information than the original coefficients. From an algebraic perspective, we introduce a q-shuffle and a family q-infiltration products for non-commutative formal power series. Finally, we apply our results to generalize a theorem of Eilenberg characterizing so-called p-group languages. We show that a language is of this type if and only if it is a Boolean combination of specific languages defined through q-binomial coefficients seen as polynomials over $\mathbb{F}_p$.

翻译：高斯二项式系数是整数二项式系数的q-模拟。另一方面，二项式系数已被推广至有限词（即有限生成自由幺半群的元素）。本文通过引入词的二项式系数的q-模拟，将这两个概念结合起来。我们研究其基本性质，例如将经典公式（如q-范德蒙恒等式和Manvel等人的恒等式）推广到我们的框架中。由此可得所考虑词的结构信息：这些词的二项式系数的q-变形蕴含比原始系数更丰富的信息。从代数视角出发，我们引入了非交换形式幂级数的q-洗牌积和一族q-渗透积。最后，我们应用所得结果推广了刻画所谓p-群语言的Eilenberg定理。我们证明：一种语言属于此类当且仅当它是通过视为$\mathbb{F}_p$上多项式的q-二项式系数定义的特定语言的布尔组合。