A049998

[Graeme McRae]

----- Original Message ----- 
From: "Don Reble" <djr at nk.ca>
To: "Seqfan" <seqfan at ext.jussieu.fr>
Sent: Wednesday, December 14, 2005 6:00 PM
Subject: Re: A049998

>First prove these identities:
>
>   F(a+1) * F(a-1) - F(a) * F(a) = (-1)^a

This is a consequence of d'Ocagne's identity: F(m) F(n+1) - F(m+1) F(n) = 
(-1)^n F(m-n)
with m=n+1=a.

> F(a+1) * F(b-1) - F(a-1) * F(b+1)
>    = + (-1)^b F(a-b), if a>b
>    = - (-1)^a F(b-a), if a<b

Since F(-a) = -(-1)^a F(a), both of these are equivalent regardless of the 
relative sizes of a and b, so I'll just prove one of them.

This identity follows from using d'Ocagne's identity twice.

First, F(a+1) F(b-1) - F(a) F(b) = (-1)^b F(a-b+1)
from d'Ocagne with m=a and n+1=b

Second, F(a) F(b) - F(a-1) F(b+1) = -(-1)^b F(a-b-1)
from d'Ocagne with m=a-1 and n=b

Adding these two together,
F(a+1) F(b-1) - F(a-1) F(b+1) = (-1)^b (F(a-b+1)-F(a-b-1)) = (-1)^b F(a-b)