q-Parikh Matrices and q-deformed binomial coefficients of words

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{g}ecio{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 (pₙ)n≥ 0 of prefixes of an infinite word, we show that the polynomial sequence pₙzq converges to a formal series. We present links with additive number theory and k-regular sequences. In the case of a periodic word u^ω, we generalize a result of Salomaa: the sequence uⁿzq 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^ω. 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⋯ k made of pairwise distinct symbols.