I am reading about generating functions in this reputable engineering textbook and the author uses the Fibonacci sequence as an example:
The Fibonacci sequence is defined by the initial conditions $a_0=0$, $a_{-1}=1$, and the recursion relation $a_n=a_{n-1}+a_{n-2}$ for $n>0$.
Using this definition, I am able to work through the author's example to get the correct generating function: $$G(x)=\frac{1}{1-x-x^2}$$
However, everywhere I read online states the initial conditions of the Fibonacci sequence as (0,1) and not (1,0). So I tried that instead, but I get a different result:
$$G(x)=\frac{1+x}{1-x-x^2}$$
This generating function seems to produce the Fibonacci sequence, but starting from the second number instead of the first $(1,2,3,5,8,13,...)$. Where did I go wrong? I would have thought that the sequence should be identical in both cases.
Many thanks for any help!
My calculations:
The author starts with a general expression for a generating function for a sequence $\{a_n\}$: $$G(x)=\sum_{n=0}^{\infty}a_nx^n$$
He then looks at the special case where $\{a_n\}$ is a linear recurrence of range $r$: $$a_n=\sum_{i=1}^rc_ia_{n-i}$$
and proves that this leads to: $$G(x)=\frac{g(x)}{f(x)}=\frac{\sum_{i=1}^rc_ix^i\left(a_{-i}x^{-i}+\cdots+a_{-1}x^{-1}\right)}{1-\sum_{i=1}^rc_ix^i}$$
So, for the Fibonacci sequence, I guess we have $r=2$ and $c_1=c_2=1$. So we get:
$$g(x)=a_{-1}+a_{-2}+a_{-1}x$$
Now I'm a bit confused because the subscripts don't seem to match up. In all the author's discussion, the "initial conditions" have strictly negative subscripts ($a_{-i}$ for $i=1,2,\dots,r$). But just for the Fibonacci example, we have initial conditions $a_0=0$ and $a_{-1}=1$. So I assume these are just off by 1. Therefore, substituting $a_{-1}=0$ and $a_{-2}=1$ gives the author's result above and using $a_{-2}=0$ and $a_{-1}=1$ gives my result above.