Sums of Certain Products of Fibonacci and Lucas Numbers

In Section 2 we prove a theorem involving a sum of products of Fibonacci numbers, and in Section 3 we prove the corresponding theorem for the Lucas numbers. In Section 4 we present three additional theorems, two of which involve sums of products of squares of Fibonacci and Lucas numbers. We require the following identities: n+k + n-k = 4 4 , * even, (1.5) Fn+k+Fn_k = LnFk, *odd, (1.6) Fn+k-Fn_k=F„Lk, kodd, (1.7) Fn+k-Fn_k = LnFk, £even, (1.8) 4+* + n-k = 4 4 , * even, (1.9) Ln+Jc + Ln_k=5F„Fk, £odd, (1.10) 4+*-4-* = 44> *odd, (l.ii) 4+jk 4-* = 5 4 4 , k even, (1.12) Ln-I^n = -2 = -L0, nodd, (1.13)