$$\sum _{x\ge0}^\Re f(x)= -\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(0)}{k!}$$
where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$
$$\sum _{x\ge0}^\Re f(x)=\sum _{k=1}^\infty \frac {\Delta^{k-1}f(0)}{k!} (0)_k$$
where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$
$$\sum _{x\ge0}^\Re f(x)=-\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(0)$$
$$\sum _{x\ge0}^\Re f(x)= \frac {1}{2} f(0) - \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(0)$$
Are all these definitions equal? If not, in what cases they are?
Are they equal if $f(x)$ is equal to its Newton series expansion?
