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

where $c_{k}=\int _{0}^{1}{\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}dx$

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

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


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

$$\sum _{x\ge0}^{R} f(x)=\int _{0}^{x}f(t)dt-{\frac {1}{2}}f(x)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(x)$$

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?