Introducing q-deformed binomial coefficients of words

TL;DR

This paper introduces q-deformed binomial coefficients for words, extending classical combinatorial identities and providing richer algebraic and structural insights into word languages and their properties.

Contribution

It develops a novel q-analogue of binomial coefficients for words, extending classical identities and applying these to characterize p-group languages.

Findings

Extended q-Vandermonde and Manvel's identities to words

Introduced q-shuffle and q-infiltration products for non-commutative series

Generalized Eilenberg's theorem for p-group languages using q-binomial coefficients

Abstract

Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we bring together these two notions by introducing q-analogues of binomial coefficients of words. We study their basic properties, e.g., by extending classical formulas such as the q-Vandermonde and Manvel's et al. identities to our setting. As a consequence, we get information about the structure of the considered words: these q-deformations of binomial coefficients of words contain much richer information than the original coefficients. From an algebraic perspective, we introduce a q-shuffle and a family q-infiltration products for non-commutative formal power series. Finally, we apply our results to generalize a theorem of Eilenberg characterizing so-called p-group languages. We show that a language is of this type if and only if it is a Boolean combination of specific languages defined through q-binomial coefficients seen as polynomials over $\mathbb{F}_p$.