Fibonacci and Lucas convolutions and divisibility.

摘要

Denote the Fibonacci and Lucas numbers by F0=0,F1=1, andFn=Fn-1 +Fn-2,for n≥2 and L0=2,L1=1, andLn=Ln-1+L n-2,forn≥ 2. We will first study the Fibonacci and Lucas convolutions. For example, we will find the sum k=0nFkF n-k=n+1L n-2Fn+15. Next, we will prove the following theorem.;Theorem. Let n be a nonnegative integer. Then 5n&vbm2;L1L3&cdots;L 2n+1i=0n 2n+1n-i -1n-iF 2i+1L2i+1. We will conclude this thesis with some open questions.