Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a summary of a study that I did some time. These notes on discrete Laplace transforms are incomplete. Briefly, we have
Give the sequence $(x_n)_{n \in \mathbb{N}}$, the discret direct Laplace transform (DDLT) denoted by $\ell_d\{x_n\}$ is defined for $s > 0$ by $$ \ell_d\{x_n\} = \sum_{n = 0}^{\infty}e^{-sn}x_n : = X(s) $$ Question: I have trouble finding $$ \ell_d\biggl\{{2n \choose n}\biggr\} $$ Thanks for any help and other ideas that can expand the theory.
The generating function of Catalan's numbers $y_n=\frac 1{n+1}\binom{2n}n$ is $$\sum_{n\geqslant 0 }\frac1 {n+1}\binom{2n}nx^n=\frac{1-\sqrt{1-4x}}{2x}$$
Multplying by $x$ and differentitating gives that $$\sum_{n\geqslant 0}\binom {2n}nx^n=\frac{d}{dx}\frac{1-\sqrt{1-4x}}{2}=\frac{1}{\sqrt{1-4x}}$$
One can also try to prove this directly, by noting that $$\binom{-1/2}k=(-1)^k4^{-k}\binom{2k}k$$ and that $$(1+x)^{\alpha}=\sum_{n\geqslant 0}\binom{\alpha}n x^n$$At any rate, any claim begs for a proof.