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

TL;DR

This paper introduces a q-deformation of Parikh matrices and binomial coefficients of words, generalizing classical results, exploring their properties, and linking to number theory, recurrence relations, and growth behaviors.

Contribution

It extends Parikh matrices to a q-deformed framework, revealing new identities, convergence properties, and connections to various mathematical areas.

Findings

Elements are q-deformations of binomial coefficients of words

Sequences satisfy linear recurrence relations in periodic cases

Minors of q-Parikh matrices are polynomials with natural coefficients

Abstract

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.