  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.
