$$\sum _{x\ge0}^\Re  f(x)=\int_0^x f(t) \,dt +\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(x)}{k!}$$

where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$

$$\sum _{x\ge0}^\Re f(x)=-\sum _{k=1}^\infty \frac {\Delta^{k-1}f(x)}{k!} (-x)_k$$

where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$


$$\sum _{x\ge0}^\Re f(x)=\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(x)$$

$$\sum _{x\ge0}^\Re  f(x)=\int_0^x f(t) \, dt - \frac {1}{2} f(x) + \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(x)$$

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?