Combinatorial sums of generalized Fibonacci and Lucas numbers. (Q2918616)

Given \(a,b\in \mathbb {R}\), define the generalized Fibonacci sequence \(\{G_n\}\) as \(G_1=a\), \(G_2=b\) and \(G_n=aG_{n-1}+bG_{n-2}\) for \(n\geq 2\). In the article under review, the authors prove various formulas involving \(\{G_n\}\). A typical example is NEWLINE\[NEWLINE G_{2n+3}=\frac {a^2+ab-b^2}{a}\sum _{i=0}^{n}\binom {n+i}{2i}+\frac {b}{a}G_{2n+2} NEWLINE\]NEWLINE for \(n\geq 0\), where \(a\neq 0\). They are able to deduce several interesting identities in the Fibonacci numbers and Lucas numbers upon specialization at \(a=b=1\) or \(a=2\) and \(b=1\), respectively. Most of the proofs are straightforward induction arguments.