Suppose $f(x)$ is analytic and infinitely differentiable at $x=1$.
$F(x)$ is another function such that, for all nonnegative $\alpha<1$, this infinite sum $F(\alpha)x^{\alpha}+F(α+1)x^{\alpha+1}+F(α+2)x^{\alpha+2}+F(\alpha +3)x^{\alpha +3}\dots$ converges to $f(x)$, at least around $x=1$.
I'm fairly certain this at least exists, but I could be wrong. Is there a name for the relationship between $f(x)$ and $F(x)$? Can $F(x)$ be defined in terms of the inverse Laplace transform of $f(x)$?
It's not true. Let $f(x) = 1$ and consider say $\alpha = \frac{1}{2}$. Then we get $\sum\limits_{n = 0}^\infty F(n + 1/2) x^n = \frac{1}{\sqrt x}$ around $x = 1$. But power series with center at $0$ converges in some neighbourhood of $0$, defines some analytical function, and by uniqueness theorem, as this function coincidens with $\frac{1}{\sqrt x}$ around $1$, this function is equal to it everywhere. But $\frac{1}{\sqrt x}$ isn't analytical in $0$, so can't be represented as power series centered in $0$.
I think the same argument works for any $f$ s.t. $f(1) \neq 0$.