Faster than polynomials, Slower than exponentials

metastability — Sat, 23 Aug 2008 14:47:39 -0800

Is there anything "in between" polynomials and exponentials? Does there exist a real-valued function f: R -> R such that
i) the limit (as x goes to infinity) of f(x) / a^x = 0 for all a > 1, and
ii) the limit (as x goes to infinity) of x^n / f(x) = 0 for all positive integers n.

In down to earth terms, "f(x) grows slower than any exponential, and faster than any polynomial."

This came up in a discussion about algorithms. "There are polynomial time algorithms, and exponential time algorithms. Well, is there anything in between?"

Maybe this is too technical a question for AskMe, but I'm sure the answer is known to some computer programmers, and I respect your guys knowledge on computer stuff.

Things I've tried so far:
Rephrasing the question as a statement about exponentially decaying functions, or about functions blowing up at the origin.
Taking the log of both sides.
Considering trig functions like sec(x), which has a pole at pi/2, combined with the homeomorphism between (-pi/2, pi/2) and the real line via the function arctan(x). This didn't seem to work, but I'm not entirely sure.
Randomly flipping through functional analysis textbooks for statements about the topology of the space of integrable functions. I think the function in question would have to be n-Schwartz for every n, or something like that.

What's the next polynomial in this series?

Mapes — Tue, 27 Nov 2007 07:23:17 -0800

What's the next polynomial in this series: x⁴+2x²+16, x⁸+40x⁶+1128x⁴+2560x²+65536, ? I'm doing some work in elasticity and have found an analytical solution to a tricky problem. The solution is a converging infinite series, and I've been able to work out the first two terms (with great difficulty). It may be easier to guess the additional terms than work them out. The next polynomial almost certainly has seven terms, including x¹² and 2,176,782,336 (6¹²). Any guesses as to what the other coefficients are, and what the general pattern is?

(The sum converges because the denominators are (4+x²)^5/2, (16+x²)^9/2, ... and every other term is subtracted.)

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Faster than polynomials, Slower than exponentials

What's the next polynomial in this series?