Affiliation:
1. School of Science, Tianjin University of Technology and Education, Tianjin 300222, China
Abstract
We introduce a context-free grammar
to generate Fibonacci and Lucas sequences. By applying the grammar
, we give a grammatical proof of the Binet formula. Besides, we use the grammar
to provide a unified approach to prove several binomial convolutions about Fibonacci and Lucas numbers, which were given by Hoggatt, Carlitz, and Church. Meanwhile, we also obtain some new binomial convolutions.
Funder
National Natural Science Foundation of China
Reference9 articles.
1. Some special Fibonacci and Lucas generating functions;V. E. Hoggatt;Fibonacci Quarterly,1971
2. Some classes of Fibonacci sums;L. Carlitz;Fibonacci Quarterly,1978
3. Exponential generating functions for Fibonacci identities;C. A. Church;Fibonacci Quarterly,1973
4. Context-free grammars, differential operators and formal power series;W. Y. C. Chen;Theoretical Computer Science,1993
5. Context-free grammars, permutations and increasing trees;W. Y. C. Chen;Advances in Applied Mathematics,2017