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Is there a common name for the growth rate of functions that are asymptotically on the order of $x^{cx}$, for some $c$? The term super-exponential is much too general. The factorial function grows in this way -- so would it be appropriate to say that similar real-valued functions have "factorial growth"?

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Note that $x^{cx} \in \Theta(x!)$ only for a particular choice of $c$ ($c=1$). It grows much faster (slower) than $x!$ for $c>1$ ($c<1$). – mjqxxxx Jul 17 '12 at 18:01
That's right. In a similar way, $e^x$ grows much faster than $2^x$, but they're both called exponential functions. I'm looking for a catch-all term that describes all functions that grow as fast as $x^{cx}$, for some value of $c$. – Dave Futer Jul 17 '12 at 18:07

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