$q$-Derivative Grammar

Context-free grammars, originating in computer science, are related to enumerative combinatorics through two distinct lines of development pioneered by Schützenberger and Chen, respectively.

In the framework established by Schützenberger and Delest-Schützenberger-Viennot, unambiguous grammars are translated into functional equations for ordinary generating functions.

Inspired by Rota's umbral calculus, Chen later developed a grammatical calculus by associating each context-free grammar with a formal derivative operator.

Dumont further developed this method through numerous combinatorial interpretations of grammars with finite and infinite alphabets.

Substantial progress in this direction has been achieved over the last decade.

In this paper, we introduce a q-analogue of grammatical calculus, which we call the q-derivative grammar.

We establish the basic framework of q-grammars and develop the q-grammatical calculus for computing q-exponential generating functions associated with q-grammars.

Concrete q-grammars are constructed to study q-Eulerian, q-Roselle and q-André polynomials, including their generating functions and recurrences.

This work extends the grammatical method to the q-setting and opens up new research directions.