Fibonacci and Lucas Polynomials by Context-Free Grammars

We introduce the context-free grammars \[\{a→ xa+b \ ; \ b → a\},\ \{a→ -xa+b \ ; \ b → a\} and \{a→ xa+by\ ; \ b→ a\}\] to generate Fibonacci and Lucas polynomials, Fibonacci and Lucas polynomials with negative indices, and bivariate Fibonacci and Lucas polynomials through the formal derivative operator defined with respect to these context-free grammars, respectively. In addition, we prove the m-term Honsberger-type identities for Fibonacci and Lucas polynomials by grammatical methods.