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到有限字，即有限生成自由幺半群的元素。本文通过引入字的二项式系数的q模拟，将这两个概念结合起来。我们研究了它们的基本性质，例如，通过将经典的q-Vandermonde恒等式和Manvel等人的恒等式等公式推广到我们的设定中。由此，我们获得了关于所考虑字的结构信息：这些字的二项式系数的q变形比原始系数包含丰富得多的信息。从代数角度看，我们为非交换形式幂级数引入了q洗牌和一族q渗透积。最后，我们应用我们的结果推广了Eilenberg刻画所谓p群语言的定理。我们证明，一个语言属于此类当且仅当它是通过将q二项式系数视为$\mathbb{F}_p$上的多项式而定义的特定语言的布尔组合。