An Identity Generator: Basic Commutators

Abstract

We introduce a group theoretical tool on which one can derive a family of identities from sequences that are defined by a recursive relation. As an illustration it is shown that $$\sum_{i=1}^{n-1}F_{n-i}F_i^2 ={1\over2}\sum_{i=1}^n(-1)^{n-i}(F_{2i}-F_i) ={F_{n+1}\choose2}-{F_n\choose2}, $$ where $\{F_n\}$ denotes the sequence of Fibonacci numbers.