Are $C^\infty$ Functions with all derivatives positive on [a,$\infty$),a$\gt$0 always made of exponential?

Question

Are there any $C^\infty$ real functions except the exponential family and gamma function family which has all the derivatives of same sign on an interval [a,$\infty$) with a$\gt$0 ? I speculate the function is always uses exponential as building blocks and it is unique defining property of exponential functions.

Please provide some instances otherwise.

I have not been able to find any so far.

Answer 1

Choose an arbitrary sequence $(a_n)$ with $a_n \ge 0$ and $\lim_{n\to \infty} \sqrt[n]{a_n} = 0$. Then the power series $$ f(x) = \sum_{n=0}^\infty a_n x^n $$ converges on $\Bbb R$, is a $C^\infty$ function, and $f$ and all its derivatives are positive on $[0, \infty)$.

Examples are $$ \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \ldots \\ \cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \ldots $$

Answer 2

Your function is necessarily a 'mixture' of exponentials. All you have to do is apply Bernstein's Theorem (https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions ) to $g(x)=f(-x)$. Thus $f(x)=\int e^{ax} dg(x)$ for some $g$.