An identity generator: basic commutators (Q1010670)

Summary: 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 \[ \sum_{i=1}^{n-1}F_{n-i}F_i^2 =\frac{1}{2}\sum_{i=1}^n(-1)^{n-i}(F_{2i}-F_i)=\binom {F_{n+1}}{2}-{\binom {F_n}{2}}, \] where \(\{F_n\}\) denotes the sequence of Fibonacci numbers.