Let $x$, $y$ be variables and $C$ be a constant. How do these $O$ notations compare?
$O(x^y)$ vs $O(x^{y+1})$ (with $x^{y+1} = x * x^y$)
and
$O(C^y)$ vs $O(C^{y+1})$ (with $C^{y+1} = C * C^y)$
My first thought is the first two $(O(x^y)$ and $O(x^{y+1})$ are not equal because the application of a non-constant coefficient results in different asymptotic behavior (x * $x^y$), while the latter two are equal because the application of a constant coefficient does not result in different asymptotic behavior (C * $C^y$). E.g $2 * 2^y$ vs $2^y$. Is this accurate?