Context-free grammars, differential operators and formal power series

Abstract

In this paper, we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substitution rules. We call such a set of rules a context-free grammar because these rules act like a context-free grammar in the sense of a formal language. Given a context-free grammar, we can associate each formal function with an exponential formal power series. In this way, we obtain grammatical interpretations of addition, multiplication and functional composition of formal power series. A surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation. We apply this method to obtain simple demonstrations of Faà di Bruno's formula, and some identities concerning Bell polynomials, Stirling numbers and symmetric functions. In particular, the Lagrange inversion formula has a simple grammatical representation. From this point of view, one sees that Cayley's formula on labeled trees is equivalent to the Lagrange inversion formula.