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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?

Michael Munta
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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.

J.G.
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  • Can one say that $F_m\sim\phi^m$ is not correct or it can be somehow used? – Michael Munta Mar 20 '19 at 10:07
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    @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. – J.G. Mar 20 '19 at 10:10
  • Isn't big O just giving the upper bound on the function? – Michael Munta Mar 25 '19 at 10:30
  • @MichaelMunta My mistake! I've edited my answer. – J.G. Mar 25 '19 at 10:31
  • 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? – Michael Munta Mar 27 '19 at 13:26
  • @Munta There shouldn't be an integer requirement. – J.G. Mar 27 '19 at 13:26