Deriving Binomial Convolution Formulas for Horadam Sequences via Context-Free Grammars

The Horadam sequence Hn(a,b;p,q) unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials (v1,v2,v3) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices kn through a parameterized grammar Gk. Using the modified grammar Gk˜, we derive convolution formulas involving the weighting term (−q)n−i. Furthermore, applying the proposed framework to (p,q)-Fibonacci and (p,q)-Lucas sequences, we derive explicit convolution formulas with parameters (p,q). The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers.