This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges.
The question: for which real-valued infinitely differentiable function $f$ defined everywhere on the real line, such that for any $x$ we have $f^{(0)}(x)+f^{(-1)}(x)+f^{(1)}(x)+f^{(-2)}(x)+f^{(2)}(x)+\ldots$ converge. Where $f^{(0)}(x)=f(x)$ and for $n>0$ then $f^{(n)}(x)=\frac{d}{dx}f^{(n-1)}(x)$ and $f^{(-n)}(x)=\int\limits_{0}^{x}f^{(-(n-1))}(t)dt$.
Clearly any polynomials is a solution thanks to finite cut off in the derivative side. Other than that, if the series converge uniformly on any nonzero compact interval, a bit of derivation (though I'm still not sure I could reorder the terms) will show that $s^{\prime}(x)=s(x)$ where $s(x)$ is the result of the series, giving us the possibility that $s(x)=ce^{x}$; I have not tried, but I presume that with some work we will be able to find that the only way to get the exponential function to be a sum is if $f$ is a polynomial. In other case the sum must not converge uniformly. By Dini's theorem, this means this series' terms must change sign infinitely often.
Anyone got any ideas on how to tackle this? Thank you.