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 Eğecioğlu 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 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 (pnz)q converges to a formal series. We present links with additive number theory and k-regular sequences. In the case of a periodic word uω, we generalize a result of Salomaa: the sequence (unz)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 uω. 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.