We have introduced a q-deformation, i.e., a polynomial in q with natural coefficients, of the binomial coefficient of two finite words u and v counting the number of occurrences of v as a subword of u. In this paper, we examine the q-deformation of Parikh matrices as introduced by E\u{g}ecio\u{g}lu in 2004. Many classical results concerning Parikh matrices generalize to this new framework: Our first important observation is that the elements of such a matrix are in fact q-deformations of binomial coefficients of words. We also study their inverses and as an application, we obtain new identities about q-binomials. For a finite word z and for the sequence $(p_n)_{n\ge 0}$ of prefixes of an infinite word, we show that the polynomial sequence $\binom{p_n}{z}_q$ converges to a formal series. We present links with additive number theory and k-regular sequences. In the case of a periodic word $u^\omega$, we generalize a result of Salomaa: the sequence $\binom{u^n}{z}_q$ satisfies a linear recurrence relation with polynomial coefficients. Related to the theory of integer partition, we describe the growth and the zero set of the coefficients of the series associated with $u^\omega$. Finally, we show that the minors of a q-Parikh matrix are polynomials with natural coefficients and consider a generalization of Cauchy's inequality. We also compare q-Parikh matrices associated with an arbitrary word with those associated with a canonical word $12\cdots k$ made of pairwise distinct symbols.

翻译：我们引入了两个有限词u和v的二项式系数的q-变形（即具有自然系数的关于q的多项式），该系数统计v作为u的子词出现的次数。本文研究了2004年Eğecioğlu引入的Parikh矩阵的q-变形。关于Parikh矩阵的许多经典结论可推广至这一新框架：首要重要发现是，此类矩阵的元素实际上是词二项式系数的q-变形。我们还研究了其逆矩阵，并作为应用获得了关于q-二项式的新恒等式。对于有限词z及无限词的前缀序列$(p_n)_{n\ge 0}$，我们证明多项式序列$\binom{p_n}{z}_q$收敛于一个形式级数。我们揭示了与加法数论和k-正则序列的联系。对于周期词$u^\omega$，我们推广了Salomaa的一个结果：序列$\binom{u^n}{z}_q$满足一个具有多项式系数的线性递推关系。结合整数分拆理论，我们描述了与$u^\omega$关联的级数系数的增长特性及零点集。最后，我们证明q-Parikh矩阵的子式是具有自然系数的多项式，并考虑了柯西不等式的一个推广。我们还比较了任意词关联的q-Parikh矩阵与由两两互异符号构成的标准词$12\cdots k$关联的q-Parikh矩阵。