A generating function is a useful encoding of a sequence $\{a_n\}_{n=1}^\infty$. That is, given a generating function $A(x)$ we can find any $a_n$ by taking derivatives. Is there a continuous analogue for this? So given a function $f(x)$ for $x \in \mathbb{R}$ can we associate some generating function for functions, $F_f(x)$ so that:
$$D^{r}|_{0}F_f(x) = f(r)$$
Where $D^{r}$ is the $r^{th}$ fractional derivative? If anyone knows related constructions let me know (replacing the $D$ operator with something else).
The generating function is also known as Z transform in the realm of probability theory. The continuous analog is the Laplace transform. Given a discrete random variable $M$, characterized by probabilities $\{p_n\}_{n=0}^\infty$, the Z transform is given by $$E[z^M]=M(z)=\sum_{n=0}^\infty z^n p_n$$ Similarly, given a continuous random variable $X$ with pdf $f(x)$, the Laplace transform is given by \begin{align} E[e^{-sX}] &= \int_{x=0}^\infty e^{-sx} f(x) dx \\ &= \int_{x=0}^\infty \sum_{n=0}^\infty \frac{({-sx})^n}{n!} f(x) dx \end{align}
See https://en.wikipedia.org/wiki/Laplace_transform#Probability_theory