Which relation symbol goes between $F_m$ and $\phi^m$?

I have already asked a similar question where the tilde notation was used (and context can be taken from there). Now I think that tilde is not the correct symbol to go between these two functions since tilde is defined as $$\lim\limits_{n\to \infty}\dfrac{f(n)}{g(n)} = 1$$, but the functions in question actually tend to $$\frac{1}{\sqrt{5}}$$, though I might be wrong.

So in terms of asymptotic analysis can anyone help me understand in which relation are these two functions exactly?

• You could use Big Theta ($\Theta$). Mar 20 '19 at 10:04

The usual approach is to write $$F_m\in\Theta(\phi^m)$$ or, in an abuse of notation, $$F_m=\Theta(\phi^m)$$. See here for some other common relations; $$\mathcal{O}$$ and $$\Omega$$, in particular, can be accurately used here.
• Can one say that $F_m\sim\phi^m$ is not correct or it can be somehow used? Mar 20 '19 at 10:07
• @MichaelMunta It is incorrect, because it implies a ratio of limit $1$. I can understand someone saying, "let's redefine it to not care about the exact limit", but we already have notation for that weaker statement; it's worth keeping the strong meaning of $\sim$ for when we'd like to say it.
• This might be a stupid question, but on the link you have provided where all notations are listed there is a formal definition column. I am wondering whether the constants there like $k, n_0$ etc need to be integers or they can be any real number? Mar 27 '19 at 13:26